Hello all, from a murky and moist Surrey. As Storm Alex blew in I realised that the last storm season has passed without reaching K, which I had been rather looking forward to.
I plodded through this in about an hour which isn’t really that slow for me, but there were too many “why on Earth didn’t I see that sooner?” moments to count. The most notable of which was failing to think of the right kind of Somali in 18a, resulting in a last entry most embarrassing for a kitty! There were a couple of components which were unknown to me, but nothing which hindered me unduly or left me in any serious doubt of the answer, which is how I like my obscurities served. Thanks, setter.

Definitions are underlined in the clues below. In the explanations, quoted indicators are in italics, explicit [deletions] are in square brackets, and I’ve capitalised and emboldened letters which appear in the ANSWER.
| Across | |
| 1 | Note on flying mammals incorporating first of innovative material (7) |
| BATISTE — TE (note) next to (on) BATS (flying mammals) containing (incorporating) the first letter of (first of) Innovative | |
| 5 | Married man goes round introducing current yearbook (7) |
| ALMANAC — M (married), which ALAN (man) goes round, all preceding (introducing) AC (current) | |
| 9 | One who butts in, talking of rosaceous tree (7) |
| MEDDLER — A homophone of (talking of) MEDLAR (rosaceous tree) | |
| 13 | Appallingly great van tax: it’s extortionate! (11) |
| EXTRAVAGANT — An anagram of (appallingly) GREAT VAN TAX | |
| 14 | Eminent university entertained by bad American music groups (11) |
| ILLUSTRIOUS — U (university) inside (entertained by) ILL (bad), US (American) and TRIOS (music groups) | |
| 15 | Admitted being possessed (5) |
| OWNED — A double definition | |
| 16 | South Carolina celebrity digesting Berlioz’s last piece (7) |
| SCHERZO — SC (South Carolina) + HERO (celebrity) surrounding (digesting) BerlioZ‘s last | |
| 17 | Doctor regularly quelled barmen that can be counted (9) |
| NUMERABLE — Make an anagram of (doctor) alternate letters of (regularly) qUeLlEd with BARMEN | |
| 18 | Warning describing fate of inquisitive Somali? (9,6,3,3) |
| CURIOSITY KILLED THE CAT — A cryptic definition | |
| 23 | Booked, being unforthcoming (8) |
| RESERVED — Two definitions | |
| 25 | He abandons the city man’s Asian headdress (6) |
| TURBAN — HE is removed from (abandons) T[he], plus URBAN (city) | |
| 27 | Middle East scholar taken at first with trailing plant (7) |
| ARABIST — Taken at first, with ARABIS (trailing plant) | |
| 30 | Fruit pie originally making us podgy (5) |
| PLUMP — PLUM (fruit) + Pie originally | |
| 32 | One leaves Moroccan port carrying a N American singer (7) |
| TANAGER — I (one) leaves TANG[i]ER (Moroccan port) containing (carrying) A | |
| 33 | Report of husky animal producing material for violin bows (9) |
| HORSEHAIR — A homophone of (report of) HOARSE (husky) and HARE (animal) | |
| 35 | New paperback briefly covering large variety of fruit (4,5) |
| CRAB APPLE — An anagram of (new) PAPERBACk without the last letter (briefly) around (covering) L (large) | |
| 36 | Greek hero‘s article initially edited in S America (7) |
| THESEUS — THE (article), then the first letter of (initially) Edited between (in) S (S) + US (America) | |
| 37 | Spot where builders work, so to speak (5) |
| SIGHT — This sounds like (… so to speak) SITE (where builders work) | |
| 38 | Pull fish back on the sheltered side (7) |
| LEEWARD — DRAW (pull) and EEL (fish), all reversed (back) | |
| 40 | Way to go on horseback, or walk with long steps? (6) |
| STRIDE — ST (way) + RIDE (to go on horseback) | |
| 41 | Musician married in centre of Boston — about fifty (8) |
| MINSTREL — M (married) + IN (in) + the centre of BoSTon + RE (about) + L (fifty) | |
| 44 | Regimental NCO‘s favour given to leading cloth-worker? (13-8) |
| QUARTERMASTER-SERGEANT — QUARTER (favour) next to MASTER (leading) and SERGE–ANT (cloth-worker?). The question mark is because we have to take “cloth” and “worker” separately, perhaps regarding a “serge-ant” as something that might whimsically be a cloth-worker | |
| 48 | Naughtily bend a rule? That can be tolerated (9) |
| ENDURABLE — An anagram of (naughtily) BEND A RULE | |
| 50 | Bill a continental noble rings about (7) |
| ACCOUNT — A (a) + COUNT (continental noble) goes around (rings) C (about) | |
| 53 | Operatic princess with house in US state (5) |
| IDAHO — IDA (operatic princess) with HO (house) in | |
| 54 | Clear water flowing round managed health resort (11) |
| TRANSPARENT — TRENT (water) flowing round RAN (managed) and SPA (health resort) | |
| 55 | Medallist initially whips ring with pronounced force (11) |
| PRIZEWINNER — The first letter of (initially) Whips + INNER (ring), next to (with) PRIZE, a homophone of (pronounced) PRISE (force) | |
| 56 | Tried again to pick up daughter after scripture lesson (7) |
| REHEARD — HEAR (to pick up) + D (daughter) after RE (scripture lesson) | |
| 57 | Turtle abandoning quiet tract of land (7) |
| TERRAIN — TERRA[p]IN (turtle) abandoning P (quiet) | |
| 58 | Doctor in broadcast regularly riled Lanarkshire town (7) |
| AIRDRIE — DR (doctor) in AIR (broadcast) and alternate letters of (regularly) rIlEd | |
| Down | |
| 1 | Rupees invested in wager relating to French national (6) |
| BRETON — R (rupees) inside (invested in) BET (wager) followed by ON (relating to) | |
| 2 | Giant bird with bill seen around part of UK (7) |
| TITANIC — TIT (bird) with AC (bill) seen around NI (part of UK) | |
| 3 | One defaming Cuban perhaps, ignoring tip with hesitation (9) |
| SLANDERER — [i]SLANDER (Cuban perhaps) without the first letter (ignoring tip) + ER (hesitation) | |
| 4 | English vehicle primarily serving Welshman (5) |
| EVANS — E (English) + VAN (vehicle) + the first letter of (primarily) Serving | |
| 5 | Arthropod from Irish islands I’d seen outside church (8) |
| ARACHNID — ARAN (Irish islands) + ID (I’d) around (outside) CH (church) | |
| 6 | It measures oxygen escaping from shooting star? (5) |
| METER — O (oxygen) escaping from METE[o]R (shooting star?) | |
| 7 | Toff, one breathing in ozone in African capital (7) |
| NAIROBI — NOB (toff) + I (one) taking in (breathing in) AIR (ozone). To me ozone is not air, but it’s in the dictionaries, e.g. Chambers has the definition: Loosely, fresh bracing air | |
| 8 | Man guarding only boss in honorary military rank (7-2-5) |
| COLONEL-IN-CHIEF — COLIN (man) around (guarding) LONE (only), then CHIEF (boss) | |
| 9 | Communicators collectively aimed to disperse after service (4,5) |
| MASS MEDIA — An anagram of (… to disperse) AIMED after MASS (service) | |
| 10 | Ancient city adopted by the German engraver (5) |
| DURER — UR (ancient city) inside (adopted by) DER (the, German) | |
| 11 | Play what irate motorists do perhaps when tailgated? (4,4,2,5) |
| LOOK BACK IN ANGER — The title of this play by John Osborne might also literally describe what a motorist might do when tailgated | |
| 12 | Particular deference (7) |
| RESPECT — Two definitions | |
| 19 | Excel with regard to start of this surgical procedure (7) |
| OVERTOP — OVER (with regard to) + the first letter of (start of) This + OP (surgical procedure) | |
| 20 | Engineer urges on extremely tardy kid (9) |
| YOUNGSTER — We are instructed to anagram (engineer) URGES ON with the outer letters of (extremely) TardY. I’m rubbish at anagrams at the best of times, but (this being 2020 after all) I had decided that the answer would probably start with G and “kid” would be a verb … | |
| 21 | Archbishop initially overwhelmed by male archdeacon’s goodness (7) |
| HEAVENS — Archbishop initially inside (overwhelmed by) HE (male) and VEN’S (archdeacon’s) | |
| 22 | Republican in sun hat, mostly relaxed and very hot (8) |
| TROPICAL — R (Republican) in TOPI (sun hat) followed by all but the last letter of (mostly) CALm (relaxed) | |
| 24 | Outside America, herb served at first with mother’s hot food (8,3,4) |
| SAUSAGES AND MASH — Outside USA (America) we have SAGE (herb), then Served at first + AND (with) + MA’S (mother’s) + H (hot) | |
| 26 | Man involved in drinking spree, a pub employee (9) |
| BARTENDER — ART (man) inside (involved in) BENDER (drinking spree) | |
| 28 | Device controlling flow of fuel — or choke? (8) |
| THROTTLE — A double definition | |
| 29 | European articles on country gents ultimately apt for playing down (14) |
| UNDERSTATEMENT — UN + DER (European articles) preceding (on, in a down entry) STATE (country) + MEN (gents) + the last letter of (ultimately) apT | |
| 31 | Country bumpkin‘s pulse a surgeon felt in the end (7) |
| PEASANT — PEAS (pulse) + A (a) + last letters of (… in the end) surgeoN felT | |
| 34 | Harsh call arresting Cleopatra’s killer (7) |
| RASPING — RING (call) around (arresting) ASP (Cleopatra’s killer) | |
| 39 | Necessity for game identified by river managers (9) |
| DARTBOARD — DART (river) + BOARD (managers) | |
| 42 | Alarm if one is kept ultimately for a soil loosener (9) |
| SCARIFIER — SCARE (alarm), in which IF (if) and I (one) goes inside (is kept), then the last letter of (ultimately) foR | |
| 43 | Native of Lima perhaps, salesman turning up with one in posh vehicle (8) |
| PERUVIAN — REP (salesman) reversed (turning up, in a down entry), with I (one) going in the concatenation of U (posh) and VAN (vehicle) | |
| 44 | One searching for game in macaques’ territory (7) |
| QUESTER — The answer is found in macaQUES‘ TERritory | |
| 45 | Kitchen device misplaced at store (7) |
| TOASTER — An anagram of (misplaced) AT STORE | |
| 46 | Item of footwear found in coach? (7) |
| TRAINER — Two definitions | |
| 47 | Origin of condiment, reportedly (6) |
| SOURCE — This sounds like (… reportedly) SAUCE (condiment) | |
| 49 | One adheres to WI religion — partly, for a start (5) |
| RASTA — Partly, foR A STArt | |
| 51 | Dried kernels represented by Constable, the artist? (5) |
| COPRA — COP (constable) + RA (the artist?) | |
| 52 | Governor dipping into Times — a character! (5) |
| THETA — HE (Governor) going inside (dipping into) TT times, then A (from the clue) | |
Not sure if that is good or bad, though!
Edited at 2020-10-03 11:39 am (UTC)