Solution to 20D has been deliberately left unsolved and is to be answered by a non-regular/novice commenter, with proper annotation. Those who have answered earlier in the week, please give others a chance.
Open for anyone to answer, if not solved by 1 PM.
ACROSS
1 Most annoyed with extremely obnoxious head of syndicate in summit (8) CROSSEST {CR{Ob...uS}{Sy...e}EST}
5 Gorgeous associate goes bare, intoxicated after drink (6) SUPERB {BaRE}*<=>{SUP}
10 One captivated by young lady working in office (7) MISSION {1} in {MISS}{ON}
11 See 3 Down
13 Extremely tense guys entering a club for relaxation (9) ABATEMENT {TensE}{MEN} in {A}{BAT}
14 Stylish fellow on island lied about relationships for affairs (12) INFIDELITIES {IN}{F}{I}{LIED*}{TIES}
18 A few words after fellow involved in deceit gets long term of imprisonment (4,8) LIFE SENTENCE {SENTENCE}<=>{LI{F}E}
21 One evaluating flight, spin etc in cricket match with plan to trap India's opener (4,5) TEST PILOT {TEST}{P{In..a}LOT}
23 Group going topless on excellent Hawaiian island (5) LANAI {cLAN}{A1}
DOWN
1 Observe beautiful European falling for Prince (6) COMPLY COM(-e+p)PLY
2 Torment in sob session (6) OBSESS [T]
3 & 11 What tortoise might do to take a risk (5,4,4,3) STICK ONES NECK OUT [C&DD]
4 Make a big thing of American involved in deplorable liaisons with teens (14) SENSATIONALISE {A} in {LIAISONS+TEENS}*
6 Fun clearly to host a relative (5) UNCLE [T]
7 Old erotic novel is popular (8) EXOTERIC {EX}{EROTIC}*
8 Venture by American group backing Tesla's trial of a product in final stage (4,4) BETA TEST {BET}{A}{SET<=}{T}
9 Promising band in arena after struggle (10,4) ENGAGEMENT RING {RING}<=>{ENGAGEMENT}
16 Dope, heroin and gin? Horrible habit (8) CLOTHING {CLOT}{H}{GIN*}
17 Bad defeat for one on cricket field (3,5) OFF STUMP {OFF}{STUMP}
20 Wife is winner of American beauty contest (6) ?I?S?S (Addendum - MISSUS {MISS US} - See comments)
Reference List
Associate = A, Drink = SUP, Left = L, Stylish = IN, Fellow = F, Island = I, Hotel = H, Park = P, European = E, Prince = P, American = A, Tesla = T, College = C, University = U, Student = L, Heroin = H