ECDSA: with knowledge of private key, can we make signatures with partially chosen content?
If the attacker has the ability of choosing the private key, then he can create a valid signature $(r,s)$ with a target value for $s$ for any message $m$. The attack works in the following way:
The attacker choose its target $s$, generates a random ephemeral key $k$ and computes the hash of the message $e = H(m)$.
Then the attacker computes the scalar multiplication $kG$ and takes $r$ as the x-coordinate of the result modulo $n$.
Then he computes the private key $d$ as:
$d = \frac{s - k^{-1}e}{k^{-1}r}$.
In this way $s$, when computed in the standard way: $s = k^{-1}(e+rd)$ will match the desired $s$.
In case the attacker can't choose the private key then I don't see how he can achieve his goal because in $k^{-1}(e+rd)$ he can control only $k^{-1}$ but that will pseudo-randomly influence the value of $r$.