Let $A(s) = (-\zeta'/\zeta)^{(r)}(s) = \sum_n a_n n^{-s}$, where $r\geq 0$. (We can consider $r=0$ first for simplicity.) Say I want to approximate $A(s)$ for $s=1+it$ by a finite sum - preferably a sum with a continuous truncation: $$A(s) = \sum_n a_n \eta\left(\frac{n}{x}\right) n^{-s} + \textrm{error term}$$ where $\eta$ is continuous and compactly supported. Of course we want an error term that is as small as possible. (I'd also like to make the error term explicit.)
I can see two routes:
Start with a Mellin integral $\sum_n a_n \eta\left(\frac{n}{x}\right) n^{-s} = \frac{1}{2\pi i} \int_{\sigma_0-i\infty}^{\sigma_0+i\infty} A(s+z) \frac{(e x)^z - x^z}{z^2} dz$ and shift the contour of integration to the left - but still within the zero-free region (the classical region, say). On the new contour of integration, use existing bounds on $A(s)$ (Trudgian, 2015). One obtains a useful bound, though the constants in those bounds on $A(s)$ are really not fantastic (particularly if we get too close to the edge of the zero-free region, and especially for $r>0$) and there seems to be a factor of $\log \log$ that might be spurious (coming from the part of the contour within distance $1$ of $s$).
Pull the contour all the way to the left and do a sum over zeros of $\zeta(s)$. One would expect this approach to be better than the more general approach in 1. (Here we are really using the fact that we are dealing with $(-\zeta'/\zeta)^{(r)}$: the residue of $(-\zeta'/\zeta)(s)$ at a zero $s=\rho$ is just the multiplicity of $\rho$.) The problem here is that, if we want a bound of the same $O(\cdot)$ quality as that given by option 1 (though presumably with better constants), we seem to need not just a zero-free region, but an estimate on how close zeros close to the line $\Re s = 1$ can be to each other. If two such zeros have $y$-coordinates within $o(1/\log t)$ of $t$, then it is non-obvious how to reproduce the estimates in 1. Just using standard estimates on $N(T)$ leads to losing a factor of $\log t$.
Options I can see:
2a) Perhaps some proofs of zero-free regions also give bounds on how close "bad" zeros can be to each other? (In general, of course, we do not even know that all zeros have multiplicity $1$.)
2b) Start from an expression such as $-\Re \frac{\zeta'}{\zeta}(s) = - \sum_\rho \Re \frac{1}{s-\rho} + \dotsc$ (de la Vallée-Poussin, Landau) and apply it for $s = 1 + c/\log t + i t$. The most straightforward approach here seems to introduce a spurious factor of $\log$ (in that $\Re \frac{1}{s-\rho}$ can be considerably smaller than $|1/(s-\rho)|$ as $s$ gets further away from $\rho$. I can see another, slightly less straightforward approach - namely, to use this formula to bound the number of bad zeros close to each other (in effect bootstrapping a known zero-free region to obtain a result that is in somse sense stronger but also less optimized).
Can you see other options? This may be very standard.
