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I assume you are talking about equisingular/topologically equivalent singularities (if you are talking about the analytical types, then the codimension is even higher). In that case, the relation can be computed from the embedded resolution of the singularity, as follows. This resolution consists in blowing up the point, and as long as the obtained curve plus the exceptional divisor does not have Normal Crossings, keep blowing up the points at which this fails. For each point $p$ that has to be blown up, let $m_p$ be the multiplicity of the curve at $p$; and let $f$ be the total number of non-satellite points to be blown up (a point is satellite if it is the intersection point of two exceptional components of previous blowups). Then $k=g+\sum m_p-f-1$. I think your bound follows (although some free points of multiplicity one often must be blown up in the embedded resolution).