Solution to 1A has been deliberately left unsolved and is to be answered only 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 Follower of most wanted leader of Yemen (6) L?C?E?
4 Pressure over young ones - small windows appearing on screens (3-3) POP-UPS {P}{O}{PUPS}
8 Hot wind from Africa to Southern Italy, crossing miles over a Saharan state (7) SIROCCO (+si-m)SIROCCO
9 Allow to join again to study at a top US university (7) READMIT {READ}{MIT}
12 Announced proper ceremony (4) RITE (~right)
13 Publicity fool, unleashes English to make sense (3,2) ADD UP {AD}{DUPe}
14 Gallant and fearless leader undone by rumour about First Lady (3,5) OLD WORLD {bOLD}{WOR{Lady}D}
16 Women are covering cold border violations by army men (3,5) WAR CRIME {W}{AR{C}{RIM}E} POS error
20 Time to perish or run (4) TROT {T}{ROT}
21 A cyber cell formed with potential future utility (10) RECYCLABLE*
23 Back to pressing bold and blunt point (7) OBTRUDE {T{B}O<=}{RUDE}
24 Secret desire! Most embarrassed! (7) COVERED {COVEt}{RED}
25 Fool with some secret ingredients (6) CRETIN [T]
26 Still hope endlessly (6) THOUGH THOUGHt
DOWN
1 Common folks in Los Angeles city consuming cocaine (5) LAITY {LA}{cITY}
2 Taken into team mission, possibly for decompositions (2-5) CO-OPTED {COOPTED+MISSION}=DECOMPOSITIONS [CA]
6 High ground, with bit of ethical stand (7) UNDERGO {GROUND+Et...l}*
7 Hint - I can still play! (9) SCINTILLA*
13 A form of carbon that is not needing oxygen to exist (9) ANAEROBIC {A+CARBON+IE}* Partial Indirect anagram
15 Holds itch, struggling to find a cleaning fabric (9) DISHCLOTH*
17 Cabal's exotic erotic energy (7) COTERIE {EROTIC}*{E}
19 Priest passed on gold at table (7) PLATEAU {P}{LATE}{AU}
22 Screen church parasite (5) LEECH {LEE}{CH}
Reference List
Pressure = P, Over = O, Southern = S, Italy = I, Miles = M, Publicity = AD, Women = W, Cold = C, Time = T, Bold = B, Cocaine = C, New = N, Energy = E, Priest = P, Screen = LEE, Church = CH