The solubility of ionic compounds in water is influenced by two principal factors: lattice energy and hydration energy. Lattice energy is the amount of energy that is released when the ions of an ionic compound come together to form a crystalline lattice. On the other hand, hydration energy is the amount of energy released when water molecules surround and interact with the ions of a solute.

The solubility of an ionic compound in water generally increases when its hydration energy is greater than its lattice energy. This is because the energy provided by the water molecules (hydration energy) compensates for and surpasses the energy required to break apart the ions in the crystal lattice (lattice energy).

For sodium sulphate ($\text{Na}_2\text{SO}_4$), it is highly soluble in water. This is because its hydration energy exceeds its lattice energy. The ions in sodium sulphate (Na+ and $\text{SO}_4^{2-}$) are effectively stabilized by the relatively high hydration energy resulting from water molecules surrounding these ions, which compensates for the energy needed to separate the ions from the lattice. Thus, Option A is a correct statement describing this scenario.

In the case of barium sulphate ($\text{BaSO}_4$), it is sparingly soluble in water because its lattice energy is greater than its hydration energy. The ions (Ba$^{2+}$ and $\text{SO}_4^{2-}$) form a very stable lattice which requires a lot of energy to be disrupted. The hydration energy provided by the surrounding water molecules is not sufficient to overcome this high lattice energy. Therefore, Option B correctly describes why barium sulphate is sparingly soluble.

Option C is incorrect as the lattice energy does indeed play a crucial role in determining the solubility of an ionic compound. It is one of the key factors that need to be overcome by hydration energy for dissolution to occur.

Option D is also incorrect because stating that the hydration energy of sodium sulphate is less than its lattice energy would imply that sodium sulphate is not soluble in water, which contradicts the known high solubility of the compound.

Therefore, Option A is correct for sodium sulphate, and Option B is correct for barium sulphate.