My iPhone 7 Plus won’t turn on, and I have to press the home button several times to get it on, and it is very delayed!

Try the solution for a frozen phone and if the problem persists move on to the more aggressive approaches and contact Apple Support if they do not resolve the matter.

If your iPhone, iPad, or iPod touch won't turn on or is frozen - Apple Support

Thank you for the link. This seems more related to a frozen screen. I believe the OP is having an issue with the screen turning on. Pressing the home button rapidly will eventually turn it on. Also, for me pressing the side button rapidly will eventually turn the screen on as well. Usually when I do this as soon as the screen does finally turn on then the phone will take a screenshot as if it is registering the home and side button being pressed at the same time even if they never were pressed simultaneously. One other thing I experience is sometimes the screen is black when a call is coming in. The home button will not turn the screen on. The side button will but it also will reject the phone call too. This is annoying when it is an important call that I've been waiting for. Any ideas?