The moduli space of curves of degree $d$ in $\mathbb P^2$ has dimension $\left( \begin{array}{c} d + 2 \\ 2 \end{array}\right)$. The most typical type of singularity is a simple cusp. Having a cusp at a given point is a codimension $4$ condition, so having $k$ cusps anywhere is codimension $2k$. This gives a moduli space of dimension

$\left( \begin{array}{c} d + 2 \\ 2 \end{array}\right) -2k$

of objects of geometric genus

$\left( \begin{array}{c} d -1 \\ 2 \end{array}\right)-k$

Setting $k=\left( \begin{array}{c} d -1 \\ 2 \end{array}\right)-g$, we have the equation:

$\left( \begin{array}{c} d + 2 \\ 2 \end{array}\right) +2g -2 \left( \begin{array}{c} d -1 \\ 2 \end{array}\right)  \geq 3g-3$


or

$g \leq 3 + \left( \begin{array}{c} d + 2 \\ 2 \end{array}\right) - 2\left( \begin{array}{c} d -1 \\ 2 \end{array}\right)$

which is solvable for only finitely many values of $g$, because the term on the right goes to $-\infty$ as $d$ goes to $\infty$. 


Thus, one cannot embed all curves of sufficiently high genus using only simple cusps. More complicated singularities could give larger reductions in the genus, at the price of comparatively larger reductions in the size of the moduli space. I don't think that can save you, but I don't know.