Across:
1 Advice from this old man in the nursery rhyme ("This Old Man, he played one, He played knick-knack on my thumb ... GIVE A DOG A BONE") – cater for 11 across, for instance! (cater for 'basenji' dog = GIVE A DOG A BONE),
10 Local rats, in a manner of speaking (INFORMS), similarly to more than one of those in 36 across (more than one ‘give an account’ = INFORMS),
11 Breed of dog (BASENJI) found in 6 down (‘base’ = BASE-) by Jack (-J-) in Ulster (Northern Ireland = NI = -N-I) (BASENJI),
12 Finishes garage, atrium, chai, hut (finishes/last letters of ‘garagE atriuM chaI huT’ = EMIT) and shed (EMIT),
13 Proves one’s generosity (GIVES) to those characters in 15 across (‘give’ = GIVE-) on bender (-S bend) (GIVES),
15 Put forward (GIVE) starting on all sides today (GIVE on all sides of today’s grid),
17 Permit (LET) some remote logging-in (some ‘remoTE Logging-in’ a roundabout way = tel) in a roundabout way (= reverse indicator) (tel =LET),
19 Fancy (NOTION) it (it) is served up (= reverse indicator) (it = -TI-) during the middle of the day (noon = NO-ON) (NOTION),
21 In The Zoo, part of the back (animal in the zoo + part of the back = DORSAL) door (door) is missing second lock (second letter of ‘lOck’ = o) (‘door’ without ‘o’ = DOR-) and the odd stall (the odd letter in ‘StAlL’= -SAL)? (DORSAL),
22 It’s relentless (NON-STOP) operating (-ON-) after (= position indicator) mid-morning (mid ‘morNing’ = N-) break (-STOP) (NON-STOP),
23 Took a look inside (X-RAYED) last box (last ‘boX’ = X-) with ready- (ready) -mix? (= anagram indicator) ready = -RAYED) (X-RAYED),
25 Flipping (= reversal indicator) over the top (over the top = OTT = -TTO) or (or = -RO-) initially gratuitous (initially ‘Gratuitous’ = G-) (GROTTO) for visiting dancer’s boss? (Santa’s GROTTO),
27 Rewritten (= anagram indicator) ode (ode = DOE) by one of The Rats (female rat = DOE),
29 The old capital? (old capital/money = MARK) Correct! (MARK),
30 Takes a flat (RENTS) pass in the end (‘pasS’ in the end/ last letter = -S) after (= position indicator) tear (RENT-) (RENTS),
31 Clare, for one (Clare Island = ISLE), immersed in tennis lesson (immersed in ‘tennIS LEsson’ = ISLE),
34 Worked (= anagram indicator) or dealt (or dealt = LEOTARD) with dancer’s outfit (LEOTARD),
35 Switches (= reversal indicator) financial institute (Allied Irish Bank = AIB = -BIA) for second time (second ‘tIme’ = -I-) for worker (man = NAM-) (NAMIBIA) in Africa (NAMIBIA),
36 Tell tales (GIVE AN ACCOUNT) – it’s what a teller in bank might do? (GIVE AN ACCOUNT).
Down:
2 One (I-) new (-N-) film (flick) cut (k) ('flick' without 'k' = -FLIC-) by projectionist at the centre ('projecTionist' at the centre = -T) (INFLICT) to cause trouble (INFLICT),
3 Seafront has no soft (‘seafront’ has no ‘soft’ = earn = EARN) bank (EARN),
4 Stabilised a Bandon base (‘stabilised’ abandon ‘base’ = tilisd = DISTIL) to make spirits (DISTIL),
5 Break the news (tell) briefly (l) (‘tell’ without ‘l’ = tel = -LET) to Marsh (bog) knocking over (= reverse indicator) (bog = GOB-) (GOBLET) wine glass (GOBLET),
6 Foundation (BASE) embraced by suburb as expected (embraced by ‘suburB AS Expected’ = BASE),
7 Holbein – the last (‘holbeiN’ the last = N-) drawn (= anagram indicator) cousin (cousin = -UNCIOS) (NUNCIOS) of The Ambassadors? (NUNCIOS),
8 Given a (given a) make-up (= anagram indicator) (given a = GIVE AN), model (EXAMPLE) (GIVE AN EXAMPLE) offers to provide a case in point (GIVE AN EXAMPLE),
9 Don’t hold back (GIVE ALL ONE HAS) – empty the tank! (GIVE ALL ONE HAS),
14 It’s one variety (VERSION) of vines or (vines or) bubbly (= anagram indicator) (vines or = VERSION),
16 Those supporting, for example, Elbow’s (BONES) orchestral leader (‘Orchestral’ leader = -O-) on opening night (opening ‘Night’ = -N-) in Belgium (Be = B-E-) with One Direction (South = -S) (BONES),
18 Cold (C-) starters of peppered ravioli and (starters ‘Of Peppered Ravioli And’ = -OPRA) (COPRA) and meat of the coconuts (COPRA),
20 Affirmative action (NOD) from rising (= reverse indicator) Mafia leader (Don = NOD),
21 Trail (DOG) taken in by go-go dancer (taken in by ‘go GO Dancer’ = god) is topsy-turvy (= reverse indicator) (god = DOG),
24 It’s kind of an assumption (A PRIORI) that impresarios dismiss mess (‘impresarios’ dismiss ‘mess’ = iprario = A PRIORI),
26 Leaving Cert History, for instance (TEST), on Prohibition (BAN) (TEST BAN) and outcome of The Treaty before it went nuclear? (TEST BAN nuclear treaty),
27 Gandhi isn’t part of hidden agenda (‘Gandhi’ isn’t part of ‘hidden agenda’ = ddenea = DEADEN) that’s dull (DEADEN),
28 The (the) training (= anagram indicator) (the = ETH-) by the man downstairs (Nick) mostly (k) (‘NICk’ without ‘k’ = -NIC) (ETHNIC) is to do with race (ETHNIC),
32 He’s (he’s) on the radio (= homophone indicator) (‘hes’ = ‘ease’ = EASE) in The Life of Riley (EASE),
33 What’s in The Magazine (AMMO in magazine = arms/explosives store)? A (A-) couple of Frenchmen (M+M = -MM-) starting to offend! (starting letter to ‘Offend’ = -O) (AMMO).