I think what the person is saying is that he has filed for benefits, having made less in wages for these weeks than he could have drawn in unemployment benefits. As in "partial unemployment."
What is usually required is confirmation from the employer that the person actually did work out the wages that he is reporting, earn the gross amount that he actually says he did, and then the partial claim weeks are processed.
However, it appears this person is peeved at the amount of time that this whole process is taking. Up to a month to get back-paid for these partial weeks is not uncommon. If he calls and asks the unemployment system worker, "Where's my check?" the worker may say that they have to wait for the wage information to be verified, and the employer has not yet returned the call to verify....
This will be dealt with by the unemployment system. It is NEVER the responsibility of the claimant to force the employer to comply with unemployment. If there is cooperation between the worker and the employer, the employer can actually fill out a form and send it electronically to the office, and the employee does not even have to file. But this is not mandatory. Of course the claimant is the most vitally interested person here, and is the one who is responsible for filing the claim if the employer will not, and for keeping the system reminded of their situation if they do not receive payments or at least a decision on the partial benefits in a timely manner. (They might possibly determine that the claimant was not eligible, but either way, he should hear something.)
Keep calling the unemployment system, reminding them that you have not yet received the partial unemployment for these weeks. You may want to send off a letter to your system's commissioner of Labor, possibly to your state representative, asking that they take an interest in helping you resolve this problem.
But also remember, unemployment benefits are not an income replacement program. If you couldn't make your payments and lost your house, if you had to go hungry because you didn't get paid unemployment benefits quickly, you cannot sue the system. There are some timeliness guidelines, you might mention this in your letter to the commissioner.
I gather you are now back at work for this same employer full time. Have you tried to talk to their HR department, asked what the issue is, why they have been unwilling to verify the wages?
It has been my experience that different people's definition of 'a very long time' is quite different. There are people who send in the forms and are on the phone the next day yelling "Where's my check?" Then there are people who will wait six months and quietly assume they have been declared not eligible without any official paperwork to say so. We do not know where in this continuium this OP falls.