Having continuing success entered into register (2,1,4)
I believe the answer is:
on a roll
'having continuing success' is the definition.
(I've seen this before)
'entered into register' is the wordplay.
I cannot really see how this works, but
'register' could be 'roll' (list of names) and 'roll' is found within the answer.
The remaining letters 'ona' is a valid word which might be clued in a way I don't understand.
This explanation may well be incorrect...
(Other definitions for on a roll that I've seen before include "Where toilet paper may be found" , "thus far unbeaten competitor?" , "Butter may be seen here" , "Enjoying much luck" , "continually successful" .)