The last paragraph in Hermann Karcher’s 1988 paper Embedded Minimal Surfaces Derived from Scherk’s Examples mentions the Enneper-Weierstraß representation of Jorge-Meeks k-Noids with a vertical handle. For k=3, one obtains a 3-ended minimal torus  like of finite total curvature like the Costa surface, but not embedded.

k=3

I suspect that this surface can be deformed through minimal immersions into the  Costa surface. These toroidal k-Noids exist for any k>2 and are limits of the toroidal singly periodic Scherk surfaces.

k=4

An attempt to make a 2-Noid with handle leads to a broken catenoid. Below is a 2.3-Noid.

k=2

Resources

Mathematica Notebook

PoVRay Sources