Record how old we are to dampen enthusiasm (10)
I believe the answer is:
discourage
'dampen enthusiasm' is the definition.
(to discourage is to reduce enthusiasm)
'record how old we are' is the wordplay.
I cannot quite see how this works, but
'record' could be 'disc' (synonyms) and 'disc' is found within the answer.
'old' could be 'o' (common abbreviation eg in OE for Old English) and 'o' is present in the answer.
'are' could be 'a' (the 'are' is a metric unit of area) and 'a' is found within the answer.
The remaining letters 'urge' is a valid word which might be clued in a way I don't understand.
This explanation may well be incorrect...
'to' acts as a link.
(Other definitions for discourage that I've seen before include "Cause loss of enthusiasm in (someone)" , "Try to stop" , "Put off" , "Talk out of" , "Deter; dishearten" .)