We describe the extension of a "viscous froth" model to the dynamics of a wet foam in a Hele-Shaw cell. The two-dimensional model includes the friction experienced by the soap films as they are dragged along the cell walls, while retaining accurate geometrical information. To explore the consequences of changing the liquid content in this situation, we consider a simple foam geometry known as a bubble lens: a bubble partially filling a narrow, straight channel with a single film spanning the gap between the bubble and the opposite wall. The triple vertices of this structure are decorated with Plateau borders whose area is related to the liquid fraction of the foam.We derive new expressions to allow the pressure in the Plateau borders to be calculated, and determine numerically the range of driving velocities for which the system reaches a steady state. As the liquid fraction increases, the lens moves more slowly than the spanning film and the spanning film is more greatly distorted, reducing the range of stable driving velocities. For higher velocities, the spanning film moves so quickly that it leaves the bubble behind, a situation which must be avoided in any particular application.