Encourage, for example, good one having lost energy (3,2)
I believe the answer is:
egg on
'encourage' is the definition.
(thesaurus)
'example good one having lost energy' is the wordplay.
'example' becomes 'eg'.
'good' becomes 'g' (abbreviation).
'having lost' indicates named letters should be taken away.
'energy' becomes 'e' (physics symbol as in E=mc2).
'one' with 'e' removed is 'on'.
'eg'+'g'+'on'='EGG ON'
'for' is the link.
(Other definitions for egg on that I've seen before include "Encourage (someone to do a risky thing)" , "Push" , "Press" , "Urge, impel" , "Fire" .)