Short-interval Perron bound for the second Chebyshev function ($\psi (x+y)-\psi (x)\ll y{\int }_{-T}^T|{\zeta }^{\prime }(c+it)/\zeta (c+it)|dt+x{\log }^{2}x/T$)

For $c=1+1/\log x$, $1\le y\le x$ and $2\le T\le x$, apply the truncated Perron formula with Von Mangoldt function coefficients at both endpoints on the same vertical line. The difference kernel is $((x+y{)}^s-x^s)/s={\int }_x^{x+y}{u}^{s-1}du$, of modulus $O(y)$. The usual near-integer error bound contributes $O(x{\log }^{2}x/T)$. The logarithmic derivative on this line is finite because of the Euler product.