Is there a fix to use iMac late 2013 as monitor for MacMini M1

Per Apple Document:

Use your iMac as a display with target display mode - Apple Support

"You can connect any of these older iMac models to another Mac from 2019 or earlier

for use as an external display."

No 2020 Mac or newer will work.

There is no fix, no hack, no workaround, no magic incantations that will make it work.

FWIW, I have a 2013 27" iMac and an M1 MacBook Air and can guarantee that fact.

I am not sure the two parts of your question are related.

For part A, just use the iMac as it is.

For part B, very likely no (others may post with absolute NO): Read this topic for a 2021 perspective on this: https://discussions.apple.com/thread/252271436?answerId=254300142022#254300142022

and this 2021 post by Allan Jones about software limitations:

https://discussions.apple.com/thread/252344497?answerId=254428494022#254428494022