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We construct two different families of properly Alexandrov-immersed surfacesin $\mathbb{H}^2\times \mathbb{R}$ with constant mean curvature $0<H\leq \frac1 2$, genus one and $k\geq2$ ends ($k=2$ only for one of these families). Theseends are asymptotic to vertical $H$-cylinders for $0<H<\frac 1 2$. This showsthat there is not a Schoen-type theorem for immersed surfaces with positiveconstant mean curvature in $\mathbb{H}^2\times\mathbb{R}$. These surfaces areobtained by means of a conjugate construction.

Genus one $H$-surfaces with $k$-ends in $\mathbb{H}^2\times\mathbb{R}$