nt.number theory – Moments of Number of Interval Restricted Divisors

I have previously asked the question A truncated divisor function sum
where the sum
$$S_f(x)=sum_{nleq x} min{f(x),d(n)}quad (1)$$
was of interest, and it was answered satisfactorily.

Here, I am interested in estimating the following quantity
$$S_a(x,m)=sum_{nleq x} #{d: d|n~mathrm{and}~dleq m}^a$$
so the divisors are restricted in size, or restricted to the interval $$(1,m)$$ not in “number” as in (1).

When $$a=1,$$ this is straightforward (as far as obtaining the main term), since the sum can be evaluated horizontally
$$S_1(x,m)=sum_{dleq m} lfloor x/d rfloor=left(sum_{dleq m} frac{x}{d}right)+O(m)=n log m + O(m),$$
and typically I’d be interested in relatively small values of $$m$$ in terms of $$x$$.

What about $$aneq 1$$? In particular, $$a=1/2,$$ or $$a=2,3,$$ etc. How can one estimate those sums?