How do I fix 0.5-second delay after pressing the Home Button on iPhone 8 Plus?

I would first try a restart if you have not already done so.

However it could be wear or dirt on a phone that was discontinued five years ago. Our first iPad, bought in early 2012, is still in use running old 32-bit games. However, its home button has been showing inconsistent response for about a year.

I cannot recommend your opening an iPhone for cleaning

Are you near-ish to an Apple retail store? Apple still shows that they will work on the 8 Plus † (see Repair). It does not say if they will do an evaluation in-store that does not require the expensive "fix other damage" option shown on that page.

They may be better able to say what's happening than those of us here, who can neither see nor touch your phone.