You have the relationship
z(r) = H - C^2/(2*g*r^2)
If H and C are not known, with g fixed at 9.8, then they make this into an entire family of curves. But the family varies trivially as a function of H and C. (For that matter, the entire curve is pretty basic, but I won't go there.) And plotting the entire family would make the result into a 4 dimensional thing.
But suppose you knew H and C? I'll write z as a function of all parameters, with g fixed.
z = @(r,H,C) H - C^2./(2*g*r.^2);
That now builds the value of g into z. We could plot z, for a known pair of H and C.You suggested C == 1, H = 0.35.
I added the black line to indicate the upper asymptote. As r-->0 from the right, this curve appoaches -inf, so there is a singularity at 0.
Could you overlay additional curves on top of that, showing the behavior of the curve as H varies? Well, yes. It would get complicated though, and more difficult to visualize.
If you change H, the entire curve becomes translated vertically in y.
Changing C changes only the scaling on the Y axis.
Honestly, those parameters are pretty simple in how they change the curve, but if you tried to stuff all that into one plot, it would become unwieldy and difficult to visualize. You might try picking just a few values for H, thus...
legend('H = 0.35,C = 1','H = 0.5,C = 1','H = 0.25,C = 1','H = 0.35,C = 2','H = 0.35,C = 0.5')
As you can see, this becomes complex, even for a simple set of parameter variations. Far easier to just understand how the curves vary.