Oddly drowsier with this medicine (4)
I believe the answer is:
dose
'this medicine' is the definition.
(dose is a kind of medicine)
'oddly drowsier' is the wordplay.
'oddly' means one should take alternating letters (letters in the odd-numbered positions are used).
The alternating letters of 'drowsier' are 'DOSE'.
'with' acts as a link.
(Other definitions for dose that I've seen before include "Medicate" , "some Viagra, perhaps?" , "Helping of medicine" , "Measured quantity of medicine" , "Measure of drug" .)