The decision variant of your modified flow problem is NP-complete, as evidenced by the fact that the subset sum problem can be reduced to it: For given items w_1, ..., w_n and a sum W, just create a source S connected to every item i via an edge S -> i of capacity w_i. Then connect every item i to a sink t via another edge i -> t of capacity w_i. Add an edge t -> T of capacity W. There exists a subset of items with cumulative weight W iif the S-T max-flow in the graph is W with your modifications.
That said, there is likely no algorithm that solves this problem efficiently in every case, but for instances not specifically designed to be hard, you can try an integer linear program formulation of the problem and use a general ILP solver to find a solution.
There might be a pseudopolynomial algorithm if your capacities are integers bounded by a value polynomial in the input size.