The tipsters choice has a sleep (3)
I believe the answer is:
nap
'sleep' is the definition.
(a nap is a short sleep)
'the tipsters choice has a' is the wordplay.
'the tipsters choice' becomes 'np' (I can't justify this - if you can you should believe this answer much more).
'has' is an insertion indicator.
'np' enclosing 'a' is 'NAP'.
Can you help me to learn more?
(Other definitions for nap that I've seen before include "Racing tip that professes to be a certainty" , "Snatch forty winks" , "Sleep briefly" , "Raised surface of a fabric and short sleep" , "A brief informal sleep" .)