Is the sentence "Drinking a cup of tea she felt much better" correct?
(In other words, is it possible not to use Perfect form "Having drunk .....'?)
Not as you've written it. e. a separate unit of information outside the main part of the clause.
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Not as you've written it. You need to put a comma after the gerund-participial clause in order to mark it as a supplementary adjunct, i.e. a separate unit of information outside the main part of the clause.
Do you have a reason for not wanting to use "having drunk"?