Hi

Do you mean Google Safe Search? This support article should help;

https://support.google.com/websearch/answer/510?hl=en&co=GENIE.Platform%3DiOS&oco=0

When I was going to search for something on Google a window popped up and asked if I wanted to do a safe search. I agreed and from then on when I went to Google a search called "MyPrivateSearch" came up. There's even a MyPrivateSearch.com. I turned off Safe Search in Google and it appears to be back to normal but I worry that MyPrivateSearch is still lurking back there.

It looks like a slightly spammy search tool trying to get you attention, something not connected to either Firefox or Google.

I recommend clearing your browsing history, this support article should help:

https://support.mozilla.org/en-US/kb/clear-browsing-history-firefox-ios