On average, how would range of shared centimorgans compare between first cousins who have same paternal grandmother versus first cousins with different paternal grandmothers, same paternal grandfather in both cases.
Another way you may see this relationship described is "half first cousin", meaning the cousins only share one grandparent.
The Shared cM Project is my go-to source for such information. According to their most recent data the expected range is between 236 and 704 cM, with an average of 554 cM. Note that this range is based on a small sample size of 23. It would therefore not be surprising if a small number of half first cousins fell outside this range. You nevertheless can use this range as an estimate of how much DNA half first cousins will typically share.