Does intermolecular force of attraction have anything to do with molecules having more energy in their liquid state than their solid state? If not, what is responsible for greater kinetic energy in liquids?
It is a common misconception that molecules in a liquid state categorically have more kinetic energy than molecules in a solid state. This is only true when the liquid is at a higher temperature than the solid, simply because translational kinetic energy is proportional to temperature.
In a molecular solid, the individual molecules cannot move very far, but they can still move, and if their centers of mass are moving at all, they have translational kinetic energy. The kinetic energy in these small motions is proportional to the temperature, just as is the case with the larger motions of molecules in a liquid.
So if a solid and liquid are in equilibrium at the same temperature (the melting point), both phases will have the same amount of kinetic energy (assuming we are not at such a low temperature that quantum effects come into play).
You are correct, however, that molecules in the liquid state have more energy than in the solid state and that intermolecular forces play a role. The difference in energy, though, is in the potential energy rather than the kinetic energy. When bonds form, whether intramolecular or intermolecular, there is a decrease in potential energy. In order to break those bonds, we have to increase the potential energy. When we add energy to convert a solid to a liquid (or a liquid to a gas), that energy goes to increasing the potential energy rather than the kinetic energy.
There's a form of inverse relationship between the strength of intermolecular force of attraction between molecules of a substance and the degree of kinetic energy of the substance. The higher the strength of intermolecular force between the molecules of a substance the lower the kinetic energy(K.E.). The molecules of solid have greater intermolecular force that exist between them compare to the molecules of liquid, and as such they have less kinetic energy (the energy of a body in motion). The the link below explains the concept of kinetic energy https://www.khanacademy.org/science/biology/energy-and-enzymes/the-laws-of-thermodynamics/a/types-of-energy