I have two functions $f(x, y) $ and $g(x, y) $, defined as: $$f(x, y) : R^2 \rightarrow R$$
$$g(x, y) : R^2 \rightarrow R$$
Let $(x, y) \in [a,b] \times [c, d] $ and we want to find a new function $A(x, y) $ as an integral on the region $[a,b] \times [c, d] $. $$A(x, y) = \int_{a} ^{b} \int_{c} ^{d} f(t, y)\, g(x,s) \, dt \, ds$$ Can we assume $\int_{a} ^{b} f(t, y)dt = p(t) $, then we write:
$$A(x, y) = \int_{c} ^{d} p(t) g(x,s) \, ds$$ $$A(x, y) = p(t) \int_{c} ^{d} g(x,s) \, ds $$
Or this is a mistake?
The result of a definite integral with respect to some variable won't have that variable in it. So an integral with respect to $t$ won't result in something with a $t$ in it.
That being said, integrating $\int_c^df(t,y)dt$ does give you a function of $y$, though. So you could set that to be $p(y)$. And yes, we do indeed have $\int_a^b p(y)g(x,s)ds=p(y)\int_a^b g(x,s)ds$.