There's a couple of reasons why the lakes could exist. Underground spring. Weight of the ice above it. Thermal activity etc.
The increase in melt water won't affect the lakes at that depth but it will affect the glacier further up. Glaciers have underground rivers. Any increase in melt water is obviously going to affect the upper part of the glacier.
https://www.physlink.com/education/askexperts/ae132.cfm
For most substances, increasing the pressure when a system is in equilibrium between liquid andsolid phases will increase the phase transition temperature. Water is one of a few specialsubstances for which the pressure lowers the temperature of transition. The basic reason is thatwater actually expands when it goes from the liquid to solid phase. In textbooks you will find theexplanation for these properties by using the Clapeyron-Clausius formula, but it is perhaps mostreadily explained using LeChatelier's principle.
This principle states that when a system is in equilibrium, any external changes that try to takeit out of equilibrium (like applying pressure to ice) will cause the system to adjust in a way tocounteract that change. This is a general property of what we mean by 'equilibrium' so it probablyderives more from the Second Law of Thermodynamics ('equilibrium is the state in which entropy ismaximized') than the First Law -- though perhaps a more creative individual could find a good wayto explain your question using that law too. In this case, if you increase the pressure on the icethe ice-water system wants to try to lower it again. It can do that by making itself fit into asmaller volume. But since water fills a smaller volume when it's liquid, rather than solid, it willgo to a lower melting point -- allowing more solid to become liquid.
Answered by: Brent Nelson, M.A. Physics, Ph.D. Student, UC Berkeley