Deeply felt because true, apart from odd bits (7)
I believe the answer is:
sincere
'deeply felt because true apart' is the definition.
I can't tell whether this defines the answer.
'from odd bits' is the wordplay.
'from' becomes 'since' (I've seen this in another clue).
'odd bits' becomes 're' (I can't justify this - if you can you should give a lot more credence to this answer).
'since'+'re'='SINCERE'
Can you help me to learn more?
(Other definitions for sincere that I've seen before include "Unaffected" , "'Genuine, not deceitful (7)'" , "Perfect" , "Honest, frank" , "Honest, heartfelt" .)