Can electricians come inspect electrical work in a finished house to make sure everything was done right?

Have a qualified professional electrician make the needed repairs. The local AHJ should make the inspection and your handyman should stand the expense of having the electrical repairs made to the work he did. Live and learn that a cheap bid does not mean inexpensive. When you hire a pro, you get the expertise that means these type of problems if they happen at all get fixed when they are found.