The W-PSR is a weaker version of the principle of sufficient reason. Essentially, the W-PSR merely claims that: for any contingent fact X, it is possible that X has an explanation. Alexander Pruss and Richard Gale have provided a lengthy defense of a modal cosmological argument for God's existence based on the W-PSR. A simple way of formalizing the argument might be:
1. The union of all contingent facts X is itself contingently true.
2. There is at least one possible world where a necessary being explains X.
3. If something is necessary in one possible world, it necessarily exists in all possible worlds.
4. Therefore, a necessary being exists.
(1) is fairly benign. The union of all contingent facts (X) would not be composed of any necessary facts, so it is reasonable to conclude that X is likewise contingent.
(2) points out that if X is going to have an explanation, then it is possible that this explanation be a necessary being. Apart from any defeaters, I can't think of any reason to doubt this premise.
(3) follows from the S5 axiom of modal logic, which states that if something is possibly necessary, then it is necessary. And, given the possible existence of some necessary being that explains the union of all contingent facts, we can rightfully infer that some necessary being exists.