swim have a swim go for a swim go swimming go to swim
In recent post, I discussed V and have a N, specifically argue and have an argument. In a comment, I added go for a swim. Later, I also thought of go swimming and go to swim.
In general, the first four seem to be interchangeable, but the last may have a different nuance.
I swam I had a swim I went for a swim I went swimming I went to swim
The first four entail that I did actually swim. The last doesn’t (automatically): I went (somewhere) with the intention of swimming. In fact, the third might also mean that I didn’t swim. I can think of a difference between We went-for-a-swim and We went (to the beach) for a swim(, but it was closed because of coronavirus restrictions).
Consider also:
I swam at the Olympics I had a swim at the Olympics I went for a swim at the Olympics I went swimming at the Olympics I went to swim at the Olympics
The first definitely means that I was a competitor. The fifth might mean that. The others probably mean that I was a casual swimmer. This difference probably has more to do with the requirements of swim at the Olympics, compare I sang at the Olympics. (True: I was in the massed choir for the opening ceremony.)
I haven’t been able to find go for a swim in the Cambridge Grammar of the English Language. Have a swim is a light verb construction, and go swimming and go to swim are catenative verb constructions, but what is go for a swim? It has some similarities with both, and is obviously a unit of meaning by itself; compare I went for a pizza. The indexes don’t help. I looked under go, for and swim, and the grammatical index doesn’t really help unless you already know what a construction is called (and GCEL often calls things by different names than everyone else). Maybe I’ll let serendipity guide me to the correct entry.
Never Pure and Rarely Simple 
Re “go for a …”, the occurrence that mystifies me is in the old folk song “Johnny has gone for a soldier.” It certainly doesn’t mean that Johnny is gay and has succumbed to romantic temptation.
https://en.wikipedia.org/wiki/Johnny_Has_Gone_for_a_Soldier
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I wouldn’t have remembered it without your prompting, but Sweet Polly Oliver “went for a soldier to fair London Town”. (Compare “went for a soldier in fair London Town” and “went to fair London Town for a soldier”.)
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Yes, and even in that song the protagonist is becoming a soldier (though she must disguise herself as a male in order to do so, something that wouldn’t be needed today) rather than seeking one out.
Perhaps “went as a soldier” would be how one would say it today rather than “went for a soldier,” at least for Polly. It’s not clear to me if that would apply to Johnny. In those times, did one become a soldier merely by declaring oneself one, or was there an induction procedure of some kind?
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