Shut up about a tenor being childish (8)
I believe the answer is:
immature
'childish' is the definition.
(similar in meaning)
'shut up about a tenor' is the wordplay.
'shut up' becomes 'immure' (I've seen this in other clues).
'about' indicates putting letters inside.
'tenor' becomes 't'.
'a'+'t'='at'
'immure' enclosing 'at' is 'IMMATURE'.
'being' is the link.
(Other definitions for immature that I've seen before include "Young" , "Not fully ripe or developed" , "Not yet adult or fully grown" , "Not yet fully-developed" , "behaviour of child" .)