Finding max flow in a network while optimizing distribution of extra resources

Question

How to find the maximum flow in a network where a subset of the edges must sum to a given total cost?

Consider the following example:

Given a network containing n nodes describing fuel stations, ℓ nodes for starting points S_i (each with k cars), and an endpoint node T, determine the maximum number of valid paths starting from some S_i ending at T. Each fuel station contains 1 unit of fuel by default, but we can add p_i extra fuel to each station, where the p_i sum to p.

If we represent fuel stations as two nodes ('in' and 'out' nodes with a single edge having weight p_i in between), we want to maximise flow while ensuring that the p_i sum to p.

Note: the problem needs to be solved using a reduction to maximum flow problem.

Tried using auxillary nodes to add 'additional fuel' to each station, but that means the flow represents both fuel and cars, which messes up the algorithm.