A different approach, using Cases and replacement rules, let you choose more easily which variable to keep constant:

data = Table[{x, y, x^2 + y^2}, {x, -1, 1, 1/5}, {y, -1, 1, 1/5}];
ListPlot[Cases[data, {4/5, y_, z_} :> {y, z}, All]]
ListPlot[Cases[data, {x_, 2/5, z_} :> {x, z}, All]]

Cases chooses the triples with given x or y value, and the replacement rules turns them into couples.

Thank you! That worked! But also want to plot vs. x, holding y constant. I didn't ask that, because I had expected the answer for y to generalize, but if it does, I don't see how.

BTW, why are there double brackets "[[]]" around the k? I thought functions used single brackets? Knowing this may answer my first question...

Here is an example table

In[1]:= data = Table[{x, y, x*y^2}, {x, 1, 3}, {y, 1, 4}]

Out[1]= {{{1, 1, 1}, {1, 2, 4}, {1, 3, 9}, {1, 4, 16}},
         {{2, 1, 2}, {2, 2, 8}, {2, 3, 18}, {2, 4, 32}},
         {{3, 1, 3}, {3, 2, 12}, {3, 3, 27}, {3, 4, 48}}}

Suppose you are interested in the first row. You want to "do the same thing" to each item in that row, get rid of the first number. Map is almost always the answer when you want to "do the same thing" to every item in a list. Rest drops the first item in a list and gives you the rest of the items. So "do" Rest to every list in the row.

In[2]:= Map[Rest, data[[1]]]

Out[2]= {{1, 1}, {2, 4}, {3, 9}, {4, 16}}

And plot that

ListPlot[Map[Rest, data[[1]]]]