One up on reading, maybe, digesting papers? (5)
I believe the answer is:
rider
'one up' is the definition.
(someone up on a horse)
'on reading maybe digesting papers?' is the wordplay.
'on' becomes 're' (regarding).
'reading maybe' becomes 'R' (one of the traditional three R's of education).
'digesting' indicates putting letters inside (inserted letters are eaten or ingested).
'papers?' becomes 'ID' (short for identification papers).
're'+'r'='rer'
'rer' going around 'id' is 'RIDER'.
(Other definitions for rider that I've seen before include "supported by an Arab, perhaps" , "Supplementary clause" , "H. - Haggard, English novelist (She, etc)" , "Horseman or supplementary jury verdict" , "Addition to verdict" .)