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Quality fish caught for the fryer (4)

Monk(fish)

Elmo knew about eccentric pianist colleague (4)

anag of elMO KNew

Jazz pianist Elmo Hope was a contemporary of Monk.
Eccentric serves as anagrind plus descriptor of both musicians (esp. TSM).

Well-rounded or fishy jazz pianist? (10,4)

Sphere being his middle name

Plonker featured in US jazz mags (10,4)

'Round midnight, barman, though exhausted, beheaded old depraved hermit (10, 4, good buddy)

T/H + (f)ELONIOUS + MONK

depraved and hermit are in Chambers as archaic for felonious and monk. Barman = composer is a pretty common device, I think.

https://www.youtube.com/watch?v=zre0u5XyNfY

Minton's outstanding newfangled keyboardist, primarily (4)

Minton's was a famous Harlem jazz club.
See famous photo at https://www.loc.gov/resource/music.musgottlieb-06201-001/

Tweak #37

Cash being tight, borrow grand from brother (4)

Mon(ey)k

Self-righteous brother? (4)

Celebrated pianist played on "Kimono My House", Ron Mael firing Mary and me(10, 4)

Tuck is possibly merry on narcotic kava roots (4)