Monkey is second in running stream (8)
I believe the answer is:
marmoset
'monkey' is the definition.
(marmoset is a kind of monkey)
'second in running stream' is the wordplay.
'second' becomes 'mo' (both are short periods of time).
'in' is an insertion indicator.
'running' indicates an anagram.
'stream' with letters rearranged gives 'marset'.
'mo' inserted inside 'marset' is 'MARMOSET'.
'is' acts as a link.
(Other definitions for marmoset that I've seen before include "Take me to arms about bushy-tailed monkey of tropical America" , "rather small creature" , "Small bushy-tailed monkey" , "Primate" , "Small New World monkey" .)