These surface probably don’t appear in the literature. They are not embedded, have high degree of the Gauß map, and their construction doesn’t pose any interesting challenges. The Enneper-Weierstraß representation is a straightforward modification of the one for the Jorge Meeks k-Noids:
There even is an additional parameter r that squeezes the surface. I have, unfortunately, a weakness for the baroque ornaments these Enneper ends provide.