The one with the shirt-like titfer? (4)
I believe the answer is:
that
'shirt-like titfer?' is the definition.
I can't tell whether this definition defines the answer.
'the one with the' is the wordplay.
'the one' becomes 'hat' (I am not sure about this - if you are sure you should believe this answer much more).
'with' is a charade indicator (letters next to each other).
'the' becomes 't' (the is pronounced as a 't' sound in some dialects).
'hat' put after 't' is 'THAT'.
Can you help me to learn more?
(Other definitions for that that I've seen before include "Take ..., English pop band" , "Yon" , "See 5" , "Not 8 Ac but the other" , "The (one) there" .)