The person you're seeing may have a heart of stone! (4)
I believe the answer is:
date
'the person' is the definition.
The answer is an example.
'you're seeing may have a heart of stone' is the wordplay.
'you're seeing' becomes 'dat' (I can't justify this - if you can you should give a lot more credence to this answer).
'may have' says to put letters next to each other.
'a heart of' indicates the central letters.
'stone' becomes 'gem' (a gemstone).
The middle letter of 'gem' is 'e'.
'dat'+'e'='DATE'
Can you help me to learn more?
(Other definitions for date that I've seen before include "The fruit for a social engagement" , "Court" , "Sweet edible fruit" , "eg 1588" , "Particular day" .)