Across
2 Across Crackdown from scrambled purpose sins (11) → SUPPRESSION Anagram
Anagram of “purpose sins”.
8 Across Church aide, locate y scrambled (7) → ACOLYTE Anagram
Anagram of “locate y”.
9 Across Doctor housing two rings bellowed (5) → MOOED Container
OO inside MED.
10 Across Reworded versions from appears rash, strangely (11) → PARAPHRASES Anagram
Anagram of “appears rash”.
13 Across Trainers' time askew, moderation (9) → RESTRAINT Anagram
Anagram of “trainers t”.
16 Across Wildly dampened thy hopes, unrewarded (5,6) → EMPTY HANDED Anagram
Anagram of “dampened thy”.
20 Across Picture, endings of bikini, album, opera, song, large (5) → IMAGE Initialism
Last letters of: bikini, album, opera, song, large.
21 Across Note followed by wasp's weapon, abstaining from food (7) → FASTING Charade
FA + STING.
22 Across Figure parts rearranged into citrus fruits (11) → GRAPEFRUITS Anagram
Anagram of “figure parts”.
Down
1 Down Drippy, blue back my, alternately (5) → LEAKY Alternate letters
Every other letter of “blue back my”, starting with the second.
2 Down Bishop persists, concealing customer (7) → SHOPPER Hidden word
Hidden inside “bishop persists”.
3 Down Funder, a pyre strangely (5) → PAYER Anagram
Anagram of “a pyre”.
4 Down East, year rising, grain (3) → RYE Reversal
E YR reversed.
5 Down Warrior from shuffled Maria, US (7) → SAMURAI Anagram
Anagram of “Maria US”.
6 Down Restraints when gridirons shed grid (5) → IRONS Deletion
GRIDIRONS without its initial GRID.
7 Down Bare bathers from jumbled stud sin (7) → NUDISTS Anagram
Anagram of “stud sin”.
11 Down Winter coat from cores of apple, track, cards, maker, heart (5) → PARKA Middle letters
Middle letters of: apple, track, cards, maker, heart.
12 Down Slimming found in birdie tingling (7) → DIETING Hidden word
Hidden inside “birdie tingling”.
14 Down Plastic compound lurking in dusty renegade (7) → STYRENE Hidden word
Hidden inside “dusty renegade”.
15 Down Hot tipples from time of death, then expires (7) → TODDIES Charade
TOD + DIES.
17 Down Shopping center hides LA between parking and pizza, slangily (5) → PLAZA Container
LA inside PZA.
18 Down Noir novelist losing extension, Japanese stock (5) → DASHI Deletion
DASHIELL without its final ELL.
19 Down Baggy, square yard holding aggregate, briefly (5) → SAGGY Container
AGG inside SY.
21 Down Supporting, by the sound of it, four (3) → FOR Homophone
Sounds like “four”.