Partitioning a square for optimal queries
A good idea seems to be to use nonoverlapping circles in the densest circle packing in 2d. Please check the corresponding Wiki-page: https://en.wikipedia.org/wiki/Circle_packing. This way you reach up to $C = 0.907$ coverage. Assuming the scittles are uniformly distributed and $N \gg \frac{1}{S_c}$, where $S_c$ is the surface of the cricles you use, we may estimate that you'll have to pay $\frac{CN\#circles}{P} = C^2N/S_cP$ for fetching and roughly $\#circles = C/S_c$ for looking in thustly covered area. We may cover the gaps, which cover about $1-C$ of the square, by correspondingly smaller cirlces again, the cost of looking should be just as before. The cost of fetching is hard to estimate, without delving into geometry to much at this point, but i guess it may be roughly $(1-C)N/P$. Also, you may terminate one you reach $N$.

A smart adversary would start placing the scittles in the gaps. I assume it's a good idea to pick a random reference for the lattice of circle placing. Also you could randomly pick the areas to  check for scittles from the main lattice (covering $C$) and from the minor lattice (covering $1-C$).

Now we are stuck with the decission which radius to pick in $[0,R]$. I assume as large as possible under the placed constrains makes sense. There is probably space for optimization by choosing some radius $r$ so that your main lattice has little overflow over your plate.

Alternatively you may use the densest packing, but with circles overlapping. This way all the gaps vanish and you skip the cost for lookups on the secondary lattice.

Hope this input helps you somewhat.