Solution to 13D 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.
ACROSS
1 A bunch of taxis moving swiftly (5) FLEET [DD]
4 Fanzine going gaga over leftists’ leader with upper-class complaint (9) INFLUENZA {FANZINE}* over {Le...s}{U}
9 Bus stop stalker seen regularly (7) SHUTTLE {SHUT}{sTaLkEr}
10 Find loo or skip preliminaries and take a little stroll behind walls? (7) INDOORS {fIND+lOO+oR+St...l}
11 Gussalufz’s dull spear (6) IMPALE {I'M}{PALE}
12 Quite possibly, Kenya’s eradicated westbound smuggling (1,4,3) I DARE SAY [T<=]
14 No second turn for idiot (4) TWIT TWIsT
15 Sinking feeling, conclusively, during one’s breakdown (5,5) GOING UNDER {f...nG+DURING+ONE}*
18 Massive craze about Gussalufz round America (10) TREMENDOUS {TRE{ME}ND}{O}{US}
19 “Bazaar” earnestly covers this subject (4) AREA [T]
22 Spooner’s appeal, frivolous in a dopey way (8) SLEEPILY (~plea silly to sleepily)
24 Composed of extremely powerful and caustic chemical (6) PLACID {Po...uL}{ACID}
26 Sad end of mammal in calamitous ice age (7) ELEGIAC {m...aL} in {ICE+AGE}*
27 Denounce a monarchy in speech (7) ARRAIGN {A}{RAIGN}(~reign)
28 Exceptional actress, really, but careless and without guile (9) ARTLESSLY {acTreSS+REALLY}*
29 Burn evil American company! (5) SINGE {SIN}{GE}
DOWN
1 Control freak to switch staff around every now and then (7) FASCIST {To+SwItCh+StAfF}<=
2 Provide sound lacking, at its core, tonal balance (9) EQUIPOISE {EQUIP}{nOISE}
3 Traitors at the top sacrificed Britain in war to give away secrets (6) TATTLE {Tr...s}{bATTLE}
4 Butter from Netherlands and Austria essentially eliminated by Brexit, unfortunately (4) IBEX BrEXIt*
6 Behind Umbertha’s cover, crude novel hidden amidst books sold cheaply (8) UNDERCUT {Um...a}{N{CRUDE*}T}
7 Only denials and retreats (5) NOOKS {NO OKS}
8 Wasted years following a tip from bogus analyst (7) ASSAYER {YEARS}* {A}{b..uS}
13 Thick, rotating bars and barriers (10) ?O?D?L?C?S (Addendum - ROADBLOCKS {(-b)ROAD(+b)B}{LOCKS} - See comments)
16 Line delivery lands on? (9) DIRECTION [CD] (Coorrection - {DI{RE}CTION} - See comments)
17 Husband’s left, overwhelmed by penile ? This is where one can seek counsel, perhaps (8) HELPLINE {H} with {L} in {PENILE}*
21 Eagerly consumes innuendos laden with gossip, ultimately (6) SLURPS {SLUR{g...iP}S}
23 Get rid of tortuous JEE and heartless CAT! (5) EJECT {JEE*}{CaT}
25 Fancy taking a girl out to swing (4) VARY VagARY
Reference List
Upper-class = U, Second = S, Britian = B, Books = NT, Husband = H, On = RE, Left = L, Girl = G