The 2's split the input field into connected regions.
This is better expressed in terms of graphs:
think of the graph which has for nodes the the 0/1 cells and where there is an edge between two nodes if the corresponding cells are adjacent on the field, that is, if and only if a stomp of Jackson can address the two.
The problem decomposes on the connected component of this graph since no stomp of Jackson can hit two cells belonging to two different components.
Also, inside one single connected component, we can always reduce the number of lighten cells down to at most 1.
Can we always get down to 0?

good question: Can you characterize when it is possible to get down to 0?

Answering the above delivers comprehension about this problem.