From the puzzle’s instructions:
Clues are of two types. From the answer to each down clue one ‘C’ has been omitted before entry (even if the full answer contains more than one ‘C’); definitions in each case lead to the untreated word, and subsidiary indications are to the treated word as entered. Each of the across clues, except 23, has been doctored by the omission of one ‘C’ at some point; the full answer is to be entered in each case.
In the explanations below the ‘doctored’ words of across clues are shown in Italics. The solutions of down clues are shown with the omitted ‘C’ in parentheses.
1. Lout seen in a rap Hamlet played with open-air auditorium. AMPHITHEATRAL (Clout; hit in anag.) Azed has not included alternative candidates for the ‘doctored’ word in all of the across clues, but has done so here, provided that ‘crap’ may be understood in the modern fashion (not yet in Chambers) as an adjective. The real candidate is ‘lout’ (for ‘clout’) with ‘a rap Hamlet’ as the anagram.
11. Man-made resin, special one left by artist. SARAN (cleft; S, RA in an) This defied Watson’s comprehension until long after the puzzle was completed. ‘One’ may be doctored as ‘cone’ or as ‘once’ and is the obvious candidate, but provides nothing remotely like a cryptic indication for our solution.
12. Bishop’s hair with little left to list suddenly. SEEL (chair; see, l) Watson, having failed to notice ‘cleft’ in the previous clue, was drawn straight to ‘hair’ for ‘chair’. Much later, he noticed the same pairing used in a down solution at 29.
14. Birds messed up loo with foreign snacks passed round. TAPACOLOS (loco; anag. in tapas) Competitors wondering whether Azed might accept an entry using a doctored anagram received some encouragement here. It is notable that ‘loo’ is not the only candidate for the doctored word since ‘with’ might be doctored as ‘witch’ albeit making complete nonsense of the surface.
15. Number one rook almost in retreat, one of a series. OLEFIN (crook; n, i, felo(n), all rev.) Here, only the second of the two ‘ones’ could be doctored feasibly. However, it actually relates to Chambers’ definition of OLEFIN at its entry for ‘oleate’: ‘n any hydrocarbon of the ethylene series.’ ‘Rook’ is our ‘crook’, so to speak.
16. Sot typically almost always bottled? Illusion. MAYA (Scot; ay in Ma(c)) The first of only two of the across clues (with that for ROCK TAR) having only one word capable of consideration as the doctored word.
19. Ma’s making more racket, row that is curbed by bishop. ROARIER (Mac’s; oar, i.e., all in RR) The first of only three of the across clues (with those for AVOURE and CRAB-APPLE) to feature doctored definitions.
33. No longer coupled? Time for wining around end of union. MEINT (wincing; n in ‘Me, it!’) Azed has hit upon a teasingly apt means of indicating our solution. Quite a few solvers may have experienced some anxiety before accepting the playground phrase from so long ago. The alternative and, perhaps, safer view is that ‘wincing’ serves to indicate an anagram of ‘time’, thus giving a parsing of ‘n in anag.’
34. Among the stars, diner’s tie hanging, truly Scottish. INTERSIDEREAL (changing; anag., real) At the time of writing, (2nd January, 2 days to get an entry in) the on-line version of the puzzle still has ‘diner’s’ misprinted as ‘dine’s’. ‘Changing’ is needed in more senses than one.
4. Archetypal bigot tucked into wine, in intermediate position to shoot. HALF-(C)OCK (Alf in hock) For the benefit of younger solvers, the reference here is to the popular television character Alf Garnet played by Warren Mitchell. One cannot imagine him deigning to drink German wine, however.
5. Air navigation system cheers airmen in extremis TA(C)AN (ta, a(irme)n) Azed has devised an ‘extremely’ apt surface for this solution: Tactical air navigation system.
6. After imaginary illness the prior worsened, unhealthily engorged. HYPERTROPHI(C) (hyp, anag.) The entry in Chambers for this solution is curiously placed (with other adjectival variants) before the substantive ‘hypertrophy’, amongst those words listed under ‘hyper-’.
9. Get tame thus ... male tiger when it’s trained. RE(C)LAIM (composite anagram) The composite is formed as our solution (minus the C) plus ‘Get’. This is found to be an anagram of ‘male tiger’. The sense of ‘reclaim’ defined is listed as ‘to win from wildness’ in Chambers. Viewed as an &lit, the whole clue would define RECLAIM by way of example only.
10. Greek monk, see, immersed in English philosopher. (C)ALOYER (lo! in Ayer) The reference here is to Sir Alfred Ayer, a frequent visitor to many crossword puzzles, especially cryptics.
11. Compound mineral shattered most of lorry in violent impact. S(C)HORL-ROCK (anag. less y in shock) Read about SCHORL and related minerals here.
21. Rock and roll pioneer styled ‘Rancho’. CO(C)HRAN (anag.) ... and here to Eddie Cochran.
Across: 17. RADOME (Act; do in anag.) 23. CENTURY (The Competition Word) 24. ROCK TAR (Crack; rock1, tar) 25. SHOLOM (cash; (ca)sh, anag.) 28. OHOS (Call; soho! (rev.)) 30. AVOURE (once; our in ave) 31. CRAB-APPLE (acrid; bap in graple) 32. ASTI (Excited; asti(r))
Down: 2. MA(C)ULA (maul, a) 3. PRI(C)ED (pried) 7. EDA(C)IOUS (ou2 in anag.) 8. (C)ASCO (hidden) 13. ES(C)ARPMENT (rase (rev.), men in PT; s.v. raze1) 18. EN(C)ARPUS (supra, (fi)ne(ry), all rev.) 22. A(C)COAST (as in ‘a cot’) 26. LU(C)KIE (I in luke) 27. (C)ORONA (or, on, a) 29. (C)HAIR (hair) 30. (C)ALID (i.e. ‘a lid’)