You can't just arbitrarily add anything, Since the evolution equation for $(x,p)$ is derived from $H$.
In this case,
$$\dot{x}=\partial_pH=\frac{p}{m}+g_0(x,p)$$
$$\dot{p}=-\partial_xH=-V'(x)+g_1(x,p)$$
Since you must have,
$$dH(q,p)=\partial_qHdq+\partial_pHdp$$
$$\rightarrow -\partial_p\dot{p}=\partial_q\dot{q}$$$$\Rightarrow -\partial_p g_1(x,p)=\partial_qg_0(x,p)$$
If so you can write,
$$H=\frac{p^2}{2m}+V(x)+f(x,p)$$
with $$g_0(x,p)=\partial_p f(x,p)\ \ \&\ \ \ g_1(x,p)=-\partial_xf(x,p)$$
One can regard the tern $f(x,p)$ as the term that couples particle's velocity and its position. A specific example of such a Hamiltonian can be, Charge particle in a uniform constant EM field. So if $\vec{B}=B\hat{k}$, Then you can choose $\vec{A}=-By\hat{i}$ so that
$$H=\frac{(p_x+eBy)^2}{2m}+\frac{p_y^2}{2m}+\frac{p_z^2}{2m}+e\phi(\vec{r})$$
If you open up the first braket you get a term that will couple the velocity and position.