What's the part of speech for "having difficulty"?
Solution 1:
I am having difficulty with the test.
No: "having difficulty" is not an adjective, nor is it here a constituent. The matrix (main) verb phrase "am having difficulty with the test" is in the present progressive aspect, where the auxiliary verb "am" is predicator and the clause "having difficulty with the test" is its complement.
"Difficulty with the test" is a noun phrase serving as object of "having". The meaning is roughly the same as "I am experiencing a problem with the test".