Should he pay zakah on the profits on goods even though he has not received them because they have been recorded as a debt owed by his partners?

Firstly:

What is required in the case of zakah on trade goods is to work out the value of the goods that are prepared for sale at the end of the hijri year. If their value reaches the minimum threshold (nisaab) by itself, or when added to cash, gold or silver, then one quarter of one tenth (2.5%) of the value must be given in zakah.

If the shop is a partnership, then you must look at the share of each partner; if it reaches the minimum threshold by itself, or when added to the partner’s other wealth, then you must give zakah.

You should not look at the total wealth of all the partners, according to the majority of scholars, as has been explained in the answer to question no. 147855 .

Secondly:

The trader must work out the value of the goods that have been prepared for sale, and add to them what he owns of cash in hand, and debts owed by people which he hopes will be repaid, then from the total of all that he should give one quarter of one tenth (2.5%).

Thus zakah is based on the following equation:

The value of the things prepared for sale + cash in hand + debts which it is hoped will be repaid.

Based on that, you must give zakah on the total debt that is owed by your partners, so long as it is a debt which you hope will be repaid.

As for debt owed by one who is in financial hardship or is taking too long to repay, no zakah is to be paid on it until it is received, at which time zakah is to be paid on it only once, according to the more correct view.

And Allah knows best.