Different voltage between the solar charge controller and the battery
You need a diagram with labels to ensure that all your points are clear. As it is some words could mean several things. 

Amps  = Watts/Volts.
At full panel rated power (Wmp) the current is
~~ Watts/Volts = Wmp/Vp = 10 Watt/14.56V = 0.7 A.
You see a voltage drop of (14.56-13.86) = 0.7V .
Resistance that would cause this = V/I
= 0.7V / 0.7A = 1 Ohm.
It is "easy enough" to get 1 Ohm of resistance from cables and connections.
IF there is in fact a 0.7V drop across the wiring then thicker wires should help.

If the above incorrectly analyses the situation then you need to provide a diagram that better explains the situation.