The specific heat capacity should vary with the sugar concentration. So here's my suggestion. First find the concentration of sugar in your frozen sugar water solution. Then create a sugar water solution with that concentration of sugar.
The first step is finding the specific heat capacity of that sugar water solution. You cannot use ice cubes for this because as the ice melts, the concentration of the solution will change. So what you should do is take a hot piece of metal with a known specific heat
,like copper for example, at a temperature $T_{hot}$ and put it into the sugar water solution in the styrofoam cup at room temperature $T_{room}$. There are many ways of doing this but I would suggest pulling the metal from water that has been boiling at 373K for a reasonable amount of time so that $T_{hot} = 373K$. After putting the hot metal in the sugar water solution, monitor it's temperature until equilibrium is reached at say $T_{final}$. Then just use the fact that net heat change is (neglecting heat loss to surrounding) zero so that you have
$$m_{w}c'(T_{final} - T_{room}) + m_{metal}c_{metal} (T_{final} - T_{hot}) = 0$$
If you measure $m_w$ and $m_{metal}$, then the only unknown is the specific heat capacity of the sugar water solution $c'$.
The next step is simple. Now instead of doing the normal experiment with ice and water, you do it with frozen sugar water and sugar water solution. Both having the same sugar concentration so that when the ice melts, specific heat capacity of the liquid is unchanged. Instead of the specific heat of water, here you use the specific heat of the sugar solution that you found in the previous experiment.
Finally, if you do not have any pure sample of metal with known specific heat, then take any piece of metal lying around and do the first experiment but this time with pure water so that the unknown to be found is $c_{metal}$.
Edit: Okay, I forgot that the latent heat of fusion for your frozen sugar water is different from that of ice. So there are two unknowns now, the specific heat and the latent heat of fusion. So you need another equation. What you do is do a third experiment with the frozen sugar water, sugar water solution and hot metal with known specific heat capacity and monitor the temperature till equilibrium. That will give you your other equation.
