We show the existence in $H^2xR$ of a family of surfaces of genus $k > 0$, finite total extrinsic curvature with two catenoidal ends and one middle planar end. The proof is based on a gluing procedure which involves a compact part of a scaled Costa-Hoffman Meeks surface of $R^3$, two pieces of a catenoid of $H^2xR$ and the hyperbolic plane $H^2x{0}$ from which we have removed a disk.