If you know that corank is the cut number, then it is $[h/2]$. Each nonseparating two-sided circle is a handle. You can remove them one by one (remove the circle and patch the two holes with disks), each time reducing $h$ by $2$.
Or, even better, remove all $r$ circles at once and patch the resulting $2r$ holes with disks. You get Euler characteristic $1-h+2r\le2$.