Overwhelm pet running inside to relative (10)
I believe the answer is:
stepmother
'to relative' is the definition.
(stepmother is a kind of relative)
'overwhelm pet running inside' is the wordplay.
'overwhelm' becomes 'smother' (I've seen this in another clue).
'running' indicates an anagram (the letters run around to different places).
'inside' indicates putting letters inside.
'pet' is an anagram of 'tep'.
'smother' enclosing 'tep' is 'STEPMOTHER'.
(Other definitions for stepmother that I've seen before include "Wicked character of fairy stories" , "A relation by marriage" , "not a natural parent" , "Relative by a further marriage of one's father" , "Stereotypically wicked member of 3" .)