Some mail servers get blacklisted due to high amounts of spam coming from them. Especially those of hosts that are not aggressive in shutting down spammers. To be fair, spam often comes from hacked sites, the owners are simply unaware. The spam sites are not necessarily “dirty” sites. That much is fact, here’s my theory. Your client’s host’s mail server could on occasion be temporarily blacklisted due to high spam traffic on the same shared mail server. Blacklisted servers AFAIK are simply ignored. There’s no indication that messages are not getting through.

After the spam volume decreases, the ban is eventually lifted and mail can get out again. This is semi-automated, which explains why Google can “make some changes” to fix the problem, but it does not persist. I had a client with a similar problem, the host claimed they could not see why mail was not getting out. Fortunately, they were willing to move the account to another server at my request. AFAIK, there has not been any more problems. This may or may not be related to your issue, it’s merely anecdotal.

Changing servers is not any guarantee of being free of trouble. The real solution is to not use unreliable email for important data. It would be better for time data to be logged into a DB table and the payroll clerk could retrieve the data directly from the DB. Take email transmission out of the process. There may be a plugin that will handle this. Of course a bespoke solution could be developed.

I realize that email is your client’s preference. It’s your job as their tech consultant to show them why it’s a bad idea and how direct access to the DB would be better and more reliable. A good UI would also be more efficient than sifting through emails. Depending on their payroll system, in theory, a report could be generated that’s compatible for direct import into payroll. No need for tedious manual transposition of time data. Probably far fetched, but it’s worth looking into.

By default, WordPress sends emails through PHP, and it creates a fake From: address ([email protected]). While that was fine back in the WP2.x era, today it will usually result in emails being flagged as spam. These days, sending email reliably through PHP requires custom settings in the mail server as well as in the PHP application.

I’d suggest using a plugin that lets you send email using SMTP.