The problem is:
$$u_{xx}-u_{xy}-2u_{yy} = 0, 0<x<\infty, 0<y<\infty$$ $$u(x,0)=\phi(x),0<x<\infty$$ $$u_y(x,0)=\psi(x)$$ $$u(0,y)=0$$
I've worked out a general solution is $$ u(x,y) = \frac{1}{3}\phi(x-y)+\frac{2}{3}\phi(x+\frac{y}{2})+ \frac{2}{3}\int_{x-y}^{x+\frac{y}{2}}\psi(s) ds$$
but I'm not really sure where to go from here. I've done reflection problems for wave equations before, but the problem says that the extended functions will need to be different, because the general solution is not d'Alembert's formula. I'm not sure how to go about extending the functions in that case.
As you noted in another question, the general solution to the PDE $$ u_{xx}-u_{xy}-2u_{yy} = 0 \tag{1} $$ on the line $-\infty<x<\infty$ is $$ u(x,y)=f(x-y)+g\!\left(x+\frac{y}{2}\right), \tag{2} $$ where $f$ and $g$ are arbitrary differentiable functions.
The general solution to $(1)$ on the half-line $0<x<\infty$ with Dirichlet boundary condition $u(0,y)=0$ is similar to $(2)$: $$ u(x,y)=f(x-y)+g\!\left(x+\frac{y}{2}\right)+h(x-y), \tag{3} $$ where, in addition to the arbitrary functions$^{(*)}$ $f$ and $g$, there is a third function $h$ whose purpose is to interfere destructively with $g$ at $x=0$ in order to enforce the boundary condition $u(0,y)=0$, that is$^{(\dagger)}$, $$ g\!\left(\frac{y}{2}\right)+h(-y)=0. \tag{4} $$ Eq. $(4)$ implies $h(\xi)=-g\!\left(-\frac{\xi}{2}\right)$, or $$ h(x-y)=-g\!\left(\frac{y-x}{2}\right). \tag{5} $$
For initial conditions $u(x,0)=\phi(x)$ and $u_y(x,0)=\psi(x)$, the solution presented in the question has the form $(2)$, with \begin{align} f(\xi)&=\frac{1}{3}\phi(\xi)-\frac{2}{3}\Psi(\xi), \tag{6} \\ g(\xi)&=\frac{2}{3}\phi(\xi)+\frac{2}{3}\Psi(\xi), \tag{7} \end{align} where $\Psi(\xi):=\int_0^{\xi}\psi(s)\,ds.$ Therefore, the solution that satisfies the boundary condition $u(0,y)=0$ is $$ u(x,y)=f(x-y)+g\!\left(x+\frac{y}{2}\right)-g\!\left(\frac{y-x}{2}\right), \tag{8} $$ with $f$ and $g$ given by $(6)$ and $(7)$, respectively.
$^{(*)}$ In the half-line case, it is convenient to define $f, g$, and $h$ on the whole real line, but with support on $[0,\infty)$.
$^{(\dagger)}$ Notice that $f(-y)=0$ if $y>0$ (see the previous footnote).