You're right that the question cannot be answered as stated, and there is no "right approach". There are infinitely many solutions of the given difference equation, so without any further assumptions there is no way to compute $y$, or any other value for that matter.
Let $p_1$ and $p_2$ be the two distinct solutions of the characteristic equation
The homogeneous solution is then given by
with arbitrary constants $A$ and $B$.
The particular solution is given by
with $q=-\frac12$. The constant $C$ is not arbitrary. It must be determined by plugging $(3)$ into the given difference equation:
The complete solution of the difference equation is given by
with $C$ given by $(4)$. The solution $y[k]$ given by $(5)$ satisfies the given difference equation for any values of $A$ and $B$. In order to determine a specific value of $y[k]$, these constants must be chosen according to some given values $y[k_1]$ and $y[k_2]$ for some values of $k_1$ and $k_2$.