How to use a supernode or supermesh with regard to this circuit?
Supernodes are used to do nodal analysis on circuits containing voltage sources. You would make a supernode from each pair of nodes that are connected by a voltage source.

Since you ask specifically about the supermesh, I won't give any more detail about supernodes. If you want more detail, you need to name all the nodes in your circuit.

Supermeshes are used to do mesh analysis on circuits containing current sources. You make a supermesh for each pair of meshes where a current source lies on a branch shared by two meshes. 

For example, in your circuit, let's call the left mesh (containing the 5 V source) #1, the upper right mesh (containing the \$2 I_x\$ CCVS) #2, and the lower right mesh (containing the 7 V source) #3. We'll define all mesh currents in the clockwise direction.

Now mesh currents \$I_2\$ and \$I_3\$ can't be determined independently because of the VCCS \$0.8 V_x\$. 

So you create a supermesh \$I_{23}\$ encompassing those two meshes, with a joining equation \$I_2 - 0.8 V_x - I_3 = 0\$. The mesh equation for the supermesh is made by following the outer perimeter of the supermesh and algebraically summing the voltage drops:

$$ -9\mathrm{V} - 2 I_x + 2 I_3 - 7 \mathrm{V} - 2 I_3 - 2 (I_3 - I_1) = 0$$