I am interested bounding the following quantity. Given fixed $k \in \mathbb{N}$, $a,b \in \mathbb{Z}$, $\sigma \in [0,1)$, and intervals $I_1,I_2 \subset \mathbb{Z}$ can we establish the bound

$$S = \sum_{x \in I_1,y \in I_2}\min\{(ax^k+by^k,q)^{1/k},q^{\sigma}\} \ll q^{\varepsilon}(|I_1||I_2|+q^{\sigma})?$$

The reason this bound seems like it might reasonable is because most of the time we have that $(ax^k,q) \neq (by^k,q)$, when this is case we have that $(ax^k+by^k,q)^{1/k} \ll \min\{(x,q),(y,q)\}$. We then get that $S \ll S_1+S_2$ where $$S_1 = \sum_{x \in I_1,y \in I_2}\min\{(x,q),(y,q),q^{\sigma}\} \ll q^{\varepsilon}(|I_1||I_2|+q^{\sigma}),$$ and $$S_2 = \sum_{\substack{x \in I_1,y \in I_2 \\ (ax^k,q) = (by^k,q)}}\min\{(ax^k+by^k,q)^{1/k},q^{\sigma}\}.$$ It feels like $S_2$ should be smaller than $S_1$ because of the GCD restriction, but I am having trouble seeing if this is actually the case or not.

One reason this might not be possible is if there is always a positive proportion of points $x,y$ in $I_1 \times I_2$ for which $(ax^k,q) = (by^k,q)$ and $(ax^k+by^k,q)^{1/k} \gg q^{\alpha}$ for some $\alpha > 0$, then we get the lower bound

$$S_2 \gg q^{\min\{\sigma,\alpha\}}|I_1||I_2|.$$

I am not sure if this is the case or not, I am not very familiar with these types of sums.