Estimates for the integral over minor arcs of exponential sums with special coefficients

Consider $$h(n)$$ to be an arithmetical function. Define $$\mathfrak{m}$$ as the union of the minor arcs of the form $$|\alpha-\frac{a}{q}|\leq 1/qQ$$, with $$(a,q)=1$$ and $$Q_0.

Fix $$N\geq Q$$ large. Do we know a lower bound for $$\int_{\mathfrak{m}}|\sum_{n\leq N}h(n)e(n\beta)|^2d\beta$$

for the additive function $$h(n)=\omega(n)$$, where $$\omega(n)$$ is the number of prime factors counting function?

Do we instead have an upper bound for the same quantity but for the multiplicative characteristic function of the sum of two squares?

Not an answer just a small comment: if the singular series coming from the problem of evaluating $$\sum _{n-n=0}h(n)h(n')$$ with the circle method is equal to $$\int _0^1\left |\sum_{n\leq N}h(n)e(n\beta)\right |^2d\beta$$ then you can write your minor arc contribution as $$\left (\int _0^1-\int _\mathfrak M\right )\left |\sum_{n\leq N}h(n)e(n\beta)\right |^2d\beta ,$$ so you just need to do the major arcs. (Maybe you know this already, but just in case.)