Is increasing the profit in exchange for extending the partnership period, after the agreed-upon period has expired and the original amount could not be provided, considered usury (riba)?

This partnership is permissible and falls under the category of Mudarabah (definition) (profit-sharing). The partner's share of the profit must be a common, unspecified fraction, not a fixed amount, and the agreement must not include a guarantee of the capital. If there is a profit, it is for both of you according to the agreement. If there is a loss, it is borne by the capital. As for requesting a delay in exchange for an increase, this is impermissible, as it constitutes Riba (usury). Your partner must fulfill the agreement, and you must show consideration for him and grant him a delay if he is in difficulty.