The phrase "having gone" is correct and usable in written English.
It is used to refer to something that happened in the past, even though the action has finished, For example: "Having gone through such a difficult period, she was determined to make a success of her life."
Exact(60)
Having gone behind early our heads could have easily dropped.
Having gone full circle, this is appeasement at its best.
"Having gone, I had time to settle myself," she said.
Having gone years without a new gadget, consumers were ready.
Having gone so far, where is there left to go?
Having gone to college here, I pretty much knew what to expect.
Having gone on this journey with my mum has made it all the more special".
Having gone to college is not a big deal for these characters, and provides no motivation.
Having gone back-to-back, Roughead is now talking about "a few more" flags.
Having gone foreign too early in 1997, I still want foreign diversification.
Having gone through the experience, they say, they would like to help others avoid it.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com