You already have the first few terms of the Taylor expansion of $\sec w$ about $w=0$ (yes, the choice of variable is deliberate).
Everywhere that you see a $w$ in that expansion, write $x+y^2$. Now it depends how many terms you want. But from the wording of the question, it looks as if expanding the $w^2$ term will be enough, and you won't even need the "$y^4$" term.
Alternately, and as a check, look up the formula for the Taylor expansion of a function $f(x,y)$ of two variables. You will need to evaluate some higher partial derivatives, since the first partials are $0$ at $(0,0)$. From the wording of the question, that is not what you are being asked to do. The question is asking you to "recycle" a known expansion to obtain a new one.
Here is a simpler example to illustrate the idea. Suppose you want the Taylor expansion of $\sin(xy^2)$. We could take partial derivatives, but that's doing things the hard way. Just find the Taylor expansion of $\sin w$, and everywhere that you see $w$, substitute $xy^2$.