Familiar issue beginning to encumber investigation (5)
I believe the answer is:
probe
'investigation' is the definition.
(I know that investigation can be written as probe)
'familiar issue beginning to encumber' is the wordplay.
'familiar issue' becomes 'prob' (informal or familiar abbreviation for 'problem').
'beginning to' suggests taking the first letters.
The initial letter of 'encumber' is 'e'.
'prob'+'e'='PROBE'
(Other definitions for probe that I've seen before include "Poke into" , "Examine searchingly" , "(Crime) investigation" , "Explore, examine" , "Inquiry" .)