jmc

Using long division, $$\begin{array}{cccccc}\text{multicolumn}2r{y}^3& -\frac{2}{3}{y}^2& +\frac{11}{9}y& -\frac{95}{27}& & \\ \text{cline}2-63y-2& 3y^4& -4y^3& +5y^2& -13y& 4\\ \text{multicolumn}2r3y^4& -2y^3& & & & \\ \text{cline}2-3\text{multicolumn}2r0& -2y^3& +5y^2& & & \\ \text{multicolumn}2r& -2y^3& +\frac{4}{3}{y}^2& & & \\ \text{cline}3-4\text{multicolumn}2r& 0& +\frac{11}{3}{y}^2& -13y& & \\ \text{multicolumn}2r& & +\frac{11}{3}{y}^2& -\frac{22}{9}y& & \\ \text{cline}4-5\text{multicolumn}2r& & 0& -\frac{95}{9}y& +4& \\ \text{multicolumn}2r& & & -\frac{95}{9}y& +\frac{190}{27}& \\ \text{cline}5-6\text{multicolumn}2r& & & 0& -\frac{82}{27}& \end{array}$$So the remainder is $\boxed{-\frac{82}{27}}$.