Record how old we are to dampen enthusiasm (10)
'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 located in the answer.
'old' could be 'o' (common abbreviation eg in OE for Old English) and 'o' is located in the answer.
'are' could be 'a' (the 'are' is a metric unit of area) and 'a' is present in the answer.
The remaining letters 'urge' is a valid word which might be clued in a way I don't see.
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".)