I don't see the point of adding the word "what" when the sentences will work perfectly well without it and will be much cleaner. For example, "I've been super busy this semester with taking care of the kids and doing my online accounting course." In that instance, you could also remove the word "with," making this sentence read, "I've been super busy this semester taking care of the kids and doing my online accounting course." But the second sentence reads better with the word "with." "I can't afford to take the family to a movie with the high cost of admission…"
Whenever you can make a sentence more concise and less wordy, do it.
Sigrid Macdonald is a manuscript editor and the author
of three books. Find her at http://sigridmacdonald.blogspot.com/