  The weak-field and slow-motion limit of $f(R,\mathcal{G})$ gravity is
developed up to $(v/c)^{4}$ order in a spherically symmetric background.
Considering the Taylor expansion of a general function $f$ around vanishing
values of $R$ and $\mathcal{G}$, we present general vacuum solutions up to
$(v/c)^{4}$ order for the gravitational field generated by a ball-like source.
The spatial behaviors at $(v/c)^{2}$ order are the same for $f(R,\mathcal{G})$
gravity and $f(R)$ gravity, and their corresponding real valued static
behaviors are presented and compared with the one in general relativity. The
static Yukawa-like behavior is proved to be compatible with the previous result
of the most general fourth-order theory. At $(v/c)^{4}$ order, the static
corrections to the Yukawa-like behavior for $f(R,\mathcal{G})$ gravity, $f(R)$
gravity, and the Starobinsky gravity are presented and compared with the one in
general relativity.
