Having done my homework I will go home.
What does having mean in this sentence?
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Have can be either a main verb (with several meanings, one of them is to posses something) or an auxiliary verb which is the case here. It doesn't have any meaning on its own; it is a part of grammatical construction called the perfect participle.
This construction is built with have in the -ing form + past participle. It is used to show that the first action was completed before the second.
At first I thought that it would sound more natural if the perfect participle was used with another clause in the past (not future) but I found this example published by University of Chicago Press (which I assume is a reputable publisher) and some other examples so I stand corrected.
Having done is the perfect participle and indicates a completed action.
You did your homework and now you will go home.