Can heat be added to a substance without causing the temperature of the substance to rise? If so, does this contradict the concept of heat as energy in the process of transfer because of a temperature difference?

Yes, heat can be added to a substance without causing the temperature of the substance to rise.
No, it does not contradict the concept of heat as energy in the processes of transfer because of a temperature difference.
When thermal energy (heat) added to a solid or a liquid, the temperature of the sample does not necessarily rise. Instead, the sample may change from one phase or state to another. Conversion of solid into liquid state is called fusion; conversion of liquid into gases state is called vaporization while conversion of solid directly into gaseous state is called sublimation. Thus ice melts and water boils, absorbing heat in each case without a temperature change. In the reverse processes (water freezes, steam condense), heat is released by the sample, again at a constant temperature. Therefore, heat can be added to a substance without causing the temperature of the substance to rise.
When the temperature of a body is raised, the kinetic energy and hence the velocity of molecules increases. Since the added heat does not increase the temperature of the sample results there is no increase of kinetic energy of the molecules. The added heat uses to break down the bond between the molecules. Therefore it does not contradict the concept of heat as energy in the processes of transfer because of a temperature difference.