It depends what you are trying to do: the second "what I have tried" syntax will not work at all (and none of the LIKE sequences would match your example).
But the first one:
...WHERE UniversityName LIKE '%UU1%'
should work, as it will match your example several times: UU14 and UU18. The "%" is a wildcard, meaning "any number of any characters", similar to the "*" in windows filenames, so '%UU1% matches any input string that contains UU1 in any location.
I suspect that you need to take another look at your data, as I suspect the
UniversityName
column does not contain "UU3-UU8-UU14-UU18-UU23-UU24-UU45-UU74" (or is has completely the wrong column name for the content it does have).