Is it permissible to consider the unpaid amount from the first session as participation in the second session, or should only the original amount be claimed as a delayed debt?

It is not permissible to consider the debt owed by the man as capital for a partnership with him for a second cycle. Nor is it permissible to tell someone who owes a debt, "Engage in a Mudarabah (profit-sharing partnership) with the debt you owe." This is the view of the majority of scholars. Therefore, you are not a partner with the man in the second cycle, and you are not entitled to any increase beyond the original amount he owes you. However, if the man gives you an increase without any prior condition from you, then there is no harm in accepting it.