Having sung an hymn, they went out. Could someone please tell me about this participial phrase?

How do you describe the tense implications of "having"? It has the feeling of past perfect tense, but it's obviously not.

Thank you kindly.
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Comments  (Page 2) 
If only the word "having" constitutes the perfect participle, are there others as well?
Clive Lots. Almost endless numbers.
I knew that was bad when I wrote it, but I couldn't see the way out.

Many thanks.
Cool Breeze There are different terms, and I often have no idea what native speakers are talking about when I read their posts!
Don't be shy about using the reply button.
BTW, I think the email notification is a work in progress.

Thanks for the help. My fear is that "having sung a few hymns" will just get lumped in as a participial phrase by virtue of "having" being a participle. What a gross oversimplification!

- A.

Edit. Thank you all again for your prompt and thorough attention to my problem. I was really screwed up about this thing!
Students: Are you brave enough to let our tutors analyse your pronunciation?
I have noticed that some of the terminology I am familiar with is not generally used by native speakers. No kidding.
I'm constantly amazed by the range of grammar terms that are used by non-native posters. Often, I have no idea what they are talking about.
For that reason, I find it hard to get interested in discussions about 'what is this language feature called'. Sometimes, I think learners get too interested in this and still cannot write correct sentences. However, I realize that different people learn in different ways. If it works for them, it works for them.

. . . I often have no idea what native speakers are talking about when I read their posts! And vice-versa, as noted above.

Best wishes, Clive