No offense, but you're not sharing useful (technical info) that could help someone here in these forum help you. So, suggest you turn on Debugging all the way to developer. In this case, due to the sporadic nature/behavior, you might have to leave debugging on to see if Moodle reports anything.
The other thing to check is server logs ... apache should have an error log. See anything in there on the data/times the disappearances occurred.
How long have you been hosted on LiquidWeb with that domain name? Reason asked ... in 2014 LW stopped offering shared servers to customers. Your server appears to be also serving out 2 other domains. One of those domains appears to be involve music. That might be a factor in what you are seeing.
Since none here have admin level access to your site (not asking for that access either), try this ... rather than logging on and expecting to see the site admin menu on front page ... yes, that's the way it should work ... navigate to a course. There should be a course admin menu there but also a link to Site Admin.
What happens if you try to get to the site admin area that route?
Also try direct again only use this URL
https://yourserver/moodle/admin/search.php
'spirit of sharing', Ken