Quick glance, half-heartedly, over last of jasmine tea (5)
I believe the answer is:
pekoe
'tea' is the definition.
(pekoe is a kind of tea)
'quick glance half-heartedly over last of jasmine' is the wordplay.
'quick glance' becomes 'peek'.
'half' indicates taking half.
'heartedly' becomes 'ko' (I can't justify this - if you can you should give a lot more credence to this answer).
'over' says to put letters next to each other.
'last of' says to take the final letters.
The final letter of 'jasmine' is 'e'.
'peek' halved is 'pe'.
'pe'+'ko'+'e'='PEKOE'
Can you help me to learn more?
(Other definitions for pekoe that I've seen before include "Superior grade of black tea" , "High-quality black tea" , "High quality black tea" , "Type of black tea" , "brew" .)