You do sweep line y=c-x with increasing c. If we maintain the set of wagons, you can see that each wagon can catch candies with y>=w+x. It is optimal to choose the one with wighest w.

Consider each candy falling as an interval like $$$[s - t, s + t]$$$

Claim: $$$j-th$$$ candy can be caught after $$$i-th$$$ candy if and only if $$$s_j - t_j \le s_i - t_i$$$ and $$$s_i + t_i \le s_j + t_j $$$ i.e. $$$j-th$$$ interval contains $$$i-th$$$ interval.

Now the task is to find minimum number of nested intervals to cover all intervals which is a classical Dynamic Programming problem like Longest Increasing Subsequence.