It's important to exhort one not to lose heart (6)
I believe the answer is:
urgent
'it's important' is the definition.
(I've seen this before)
'exhort one not to lose heart' is the wordplay.
'exhort' becomes 'urge' (synonyms).
'to lose heart' suggests removing the centre (I've seen 'lost heart' mean this).
'not' with its middle removed is 'nt'.
'urge'+'nt'='URGENT'
'to' is the link.
(Other definitions for urgent that I've seen before include "Pressing, burning" , "Calling for immediate attention" , "Earnest, insistent" , "High priority" , "Requiring immediate action" .)