Problem needing solution originally, let me think (3)
I believe the answer is:
sum
'problem' is the definition.
(arithmetic problem)
'solution originally let me think' is the wordplay.
'originally' indicates taking the first letters.
'let me think' becomes 'um' (expression of hesitation).
The initial letter of 'solution' is 's'.
's'+'um'='SUM'
'needing' is the link.
(Other definitions for sum that I've seen before include "Add together" , "Total; problem" , "Amount, aggregate" , "Arithmetical problem" , "Quantity obtained by addition" .)