My favorite reference for these sorts of things that straddle physics and geometry is Frankel's "The geometry of physics". He gives "Hodge's Theorem" in his chapter on harmonic forms. It's a little more general than you need, because it applies to general $p$-forms, and you only need functions ($0$-forms). So I'll specialize it to functions.
Hodge's Theorem (for functions): Let $M^n$ be a closed Riemannian manifold. Then Poisson's equation \begin{equation} \Delta \alpha = \rho \end{equation} (where $\alpha$ and $\rho$ are real-valued functions, and $\Delta$ is the Laplacian) has a solution $\alpha$ if and only if $\rho$ has mean value $0$ on $M^n$: \begin{equation} \int_M \rho\ \mathrm{vol}^n = 0. \end{equation}
So, to apply this theorem to your question, we have \begin{align} M &= \partial \mathscr{B}, \\ \alpha &= y, \\ \rho &= \mathrm{const} - \left( p_{b ; d} \hat{h}^{bd} - R_{ac} Y^{a}_{1} Y^{c}_{2} + R_{adcb} Y^{d}_{1} Y^{c}_{2} Y^{a}_{2} Y^{b}_{1} \right). \\ \end{align} [Note that this is leaves out the $p'^a p'_a$ term in (1) from the original question; that term involves derivatives of $y$ other than the Laplacian, so Hodge's theorem doesn't apply to them. But also note that Hawking & Ellis don't actually claim that it should be included in what is equal to a constant.]
As Hawking & Ellis point out, $p_{b ; d} \hat{h}^{bd}$ is a pure divergence. So you can use Stokes' theorem to transform its derivative over $\partial \mathscr{B}$ into a derivative over the boundary of $\partial \mathscr{B}$. But the boundary of a boundary is empty, so the integral has value $0$.
The theorem then states that $y$ has a solution if and only if \begin{equation} \mathrm{const}\ \int_{\partial \mathscr{B}} \mathrm{vol}^n = \int_{\partial \mathscr{B}} \left( - R_{ac} Y^{a}_{1} Y^{c}_{2} + R_{adcb} Y^{d}_{1} Y^{c}_{2} Y^{a}_{2} Y^{b}_{1} \right)\ \mathrm{vol}^n. \end{equation} Now, the volume is assumed to be nonzero, and any volume is non-negative, so the left-hand side is just the constant in question times some positive number. Thus, as Hawking & Ellis claim, there exists a $y$ such that the first four terms in the original question's (1) add up to a constant, the sign of which is determined by the integral on the right-hand side above.