Let $N\geq1$. Could we infer $$\sum_{n\leq N}\mu(n)\ll N\exp(-c\sqrt{\log N})$$from $$\sum_{n\leq N}\Lambda(n)= N+O(N\exp( -c\sqrt{\log N})$$or $$\sum_{p \leq N}1=Li(x)$$ without resorting to the Vaughan's identity (Perhaps, the Vaughan's identity does not wok. The factorization of primes seems plausible)? Here $\Lambda(n)$ is the von Mangoldt function.
Further let $N\geq1$ and any $\alpha \in \mathbb{R}$. Let $a(n)$ be the $n$th normalized Fourier coefficient of a primitive holomorphic or Maass cusp form for $SL(2,\mathbb{Z})$. Could we infer $$\sum_{n\leq N}\mu(n) a(n)e(n\alpha) \ll_f N\exp(-c\sqrt{\log N})$$from $$\sum_{n\leq N}\Lambda(n)a(n)e(n\alpha)\ll_f N\exp( -c\sqrt{\log N})$$or $$\sum_{p \leq N}a(p)e(\alpha p)\ll_f N\exp( -c\sqrt{\log N})$$ 'directly' as above? Here $$\sum_{n\leq N}\mu(n) a(n)e(n\alpha) \ll_f N\exp(-c\sqrt{\log N})$$ has been given by Fouvry and Ganguly. (c.f. Strong orthogonality between the Mobius function, additive characters, and Fourier coefficients of cusp forms)
I really wish someone to help me out, and any advice will be highly appreciated.