Across:
1 Difficulty getting head around the fact (HARD TO BELIEVE) Elvis is on the moon, for instance (HARD TO BELIEVE),
10 Spends a lot of time hanging about (ACROBAT) overseeing The Ring (ACROBAT),
11 Built (= anagram indicator) canoe (canoe = OCEAN-) for island (-I-) area (-A) (OCEANIA) in the South Pacific (OCEANIA),
12 Dig your own grave on the outskirts of (‘DIg your own graVE’ on the outskirts of = DIVE) Leap? (DIVE),
13 The odd dressy (the odd letters of ‘DrEsSy = des = D-ES) neighbour (-UK-) inside (DUKES) with noblemen (DUKES),
15 Tough (HARD) starters for all sides today (HARD on all sides of today’s grid),
17 Free (RID) part of cartridge (part of ‘cartRIDge’ = RID),
19 It’s well (OIL RIG) for those doing the boring stuff (OIL RIG),
21 Book dealers asked out (‘book dealers’ ‘asked’ out/without ‘asked’ = booler = BOLERO) to dance in Madrid (BOLERO),
22 Create (create) trouble (= anagram indicator) (create = TEAC-ER) outside hospital (-H-) (TEACHER) teaching the student nurses a lesson presumably (TEACHER),
23 It’s limited (MEAGRE) to characters involved in unwholesome agreement (characters involved in ‘unwholesoME AGREement’ = MEAGRE),
25 When (when) Dizzy (= anagram indicator) (when = N-HEW) covered record (Extended-Play record = -EP-) (NEPHEW) by one from the familial group (NEPHEW),
27 A type of communal gathering (BEE) with endless (r) drink (beer) (‘beer’ without ‘r’ = BEE),
29 Tracy left Cary Grant (‘Tracy’ left ‘Cary Grant’ = rgan = GRAN) in the old folks home perhaps (GRAN),
30 Venezuela lacks zeal (‘Venezuela’ lacks ‘zeal’ = venue = VENUE) for the game (VENUE),
31 Temporary release (BAIL) from the extremities of Lion of The Punjab, going all the way back (= reverse all indicator) (from the extremities of ‘LIon of the punjAB’ = liab = BAIL),
34 Broadcast (-AIR-) flower (flower/river -OB-) in Ulster (N-I) (NAIROBI) from the green city in the sun (NAIROBI known as “The Green City in the Sun”),
35 Sounds of boredom? (HEIGH-HO) Sing a song from first-class songbook! (HEIGH-HO),
36 Struggling to pick up what’s been said (HARD OF HEARING), as can’t get the sound right (HARD OF HEARING).
Down:
2 A (A-) final writer (final 'writeR' = -R-) to compare with (-RIVAL) (ARRIVAL) from birth (ARRIVAL),
3 Sets up (= reversal indicator) nightspot (bed = DEB-) on square (-T square) (DEBT) with borrowed capital (DEBT),
4 Outline (contour) starting off (c) (‘contour’ without ‘c’ = ontour = ON TOUR) of where the band is going (ON TOUR),
5 Journalist (E-D) admits staff (-ROD-) morale ultimately (‘moralE’ ultimately = -E-) (ERODED) is worn down (ERODED),
6 Covered up (= reverse indicator) by Thomas Alva Edison (‘thomas alvA EDIson’ = aedi = IDEA), in theory (IDEA),
7 Classic (VINTAGE) grapes grow on it (vine = VIN-E) around stub (-TAG-) (VINTAGE),
8 Just can’t see it when you close your eyes (HARD TO IMAGINE), as it’s difficult to get the picture (HARD TO IMAGINE)
9 It’s not credible (HARD TO SWALLOW) those with a sore throat find this is a problem (HARD TO SWALLOW)
14 Middle Eastern (middle ‘easTern’ = -T-) revolutionary (-CHE-) in family (kin = KI-N) (KITCHEN) of one serving 32 down? (KITCHEN is one serving ‘food’),
16 Man taken out of terminal (“man” taken out of “terminal” = teril = LITRE) has capacity to hold less than two pints (LITRE = c. 1.75 pints/less than two pints),
18 Flew off from flower bed (‘flew’ off from ‘flower bed’ = orbed = BORED) utterly jaded (BORED),
20 Find it before whizz (GEE whizz) rises (= reverse indicator), close to Kildare (close to ‘kildarE’ = -E), for example (e.g. = GE-) (GEE),
21 Some corvine birds (some ‘corviNE Birds’ = neb) around (= reverse indicator) (neb = BEN) mountain (BEN),
24 Old bean (ARABICA) – this is a person from Middle East (ARAB-) women’s organisation (Irish Countrywomen’s Association = -ICA) (ARABICA),
26 Warm weather (HEAT-) for one getting married (-HEN) (HEATHEN) to one who doesn’t have 27 down, for instance (one who doesn’t have ‘belief’ = HEATHEN),
27 Contradict a claim (BELIE-) by female (-F) (BELIEF) in Tenet (tenet/principle = BELIEF),
28 Prompt (cue) upsets (reverse indicator) (cue = EUC-) every second charmer (every second letter in ‘cHaRmEr’ = -HRE) (EUCHRE), as Trump’s playing the card game in the US? (American card game = EUCHRE),
32 All you might need to sustain yourself? (FOOD) 9 down could be said of this! (FOOD could be ‘hard to swallow’),
33 Last stand (BIER) with stodge from Gobi Desert (‘stodge’ from ‘Gobi Desert’ = bier = BIER).