Hush-hush solution
by Kegler

As indicated by the unclued phrase, TAKING PART IN A COVER-UP, twelve of the answers are replaced in
 the grid by things that cover them.  The actual grid entry is shown in brackets following the parsing of the answer.

Hush-hush grid

Across
7P(I + L)LOW [SLIP]
9AM + OS (rev)
11BED (1st letters) [QUILT]
12PAT + H
13UM + P
15ISLET (TILES anag)
16E + A + RN
18F(LO)OR [CARPET]
19EYES ("AYES" hom) [BLINDFOLD]
24CAPITOL (TOPICAL anag) [DOME]
26MIME (hidden)
28NAR + Y (RAN rev)
29NADIR (DRAIN anag)
30B + ROOM
34C + AND + Y [WRAPPER]
36V(I)A
37TAPIR ("TAPER" hom)
38BEtrayER
Down
1HEAD (hidden) [TOQUE]
2ALUM (hidden)
3bILLy
4NITS (SNIT anag)
5W + ALL [PAPER]
6RO(T + T)ED
8P + ILAFS (FAILS anag)
10WI(N)DOW [SHUTTER]
14PRIOR (2 defs)
17AL + DA
20D(EM)O (ME rev)
21OMARR (ARMOR anag)
22P + AN [LID]
23PEA + K [SNOW]
25MY(O) + PIC
27M(I + LIE)U
30B(R)AN
31ME + RV
32EVER (VEER anag)
33INFIELD (INFIDEL anag) [TARP]
35P(R)O