I thought Freeplay and BayGen are essentially the same company, just something to do with legalities and names and that one is the commercial manufacturing part and the other is the design part that created the spring-powered equipment as part of a 3rd-World assistance project.
Anyway, the Freeplay 2020, IIRC, has no solar panel. The LEDs run straight off the spring. The 10min run time is how long it takes the spring to fully unwind from fully wound position. The LEDs don't have any perceptible flicker because the unwinding spring keeps the generator speed constant and what little fluctuation is further smoothed by a small capacitor hooked up between the output terminals of the generator. But between the generator and the LEDs, there is no proper power storage unit. The incandescent bulb is powered by a completely separate battery system that is not rechargeable via the spring-generator system. The spring unwinds slowly because of a governor system (those gears you see in the flashlight or radio).
The systems that I am looking in to modifying are the Freeplay S360 radio and the Freeplay FPSL flashlight. They are out of production, but I bought mine a while back when my local electronics/hobby stockist was having them on sale. The FPSL looks like it may be the easiest to mod since the spring-generator and batteries are connected, and there is enough space to put in extra stuff (like a solar panel, banks of capacitors, regulators, etc.) The S360 has a solar panel already installed but has very little space for anything extra. The difficulty with the mod is that the batteries are 2.4V and that the spring-generator may be set up to deliver only 3V to charge them. And also, opening up anything with a wound spring in it is a little risky.
BTW, I'm not sure about the newer models being offered by Freeplay. The Sherpa, Sentinel, and Outrider seem to have discarded the spring. Instead, the user has to wind at a constant and optimal rate, which is shown by a couple of indicator lights. A step backwards, IMHO.
Also, there are quite a few forumites who have modded a standard, hand-cranked incandescent flashlight which doesn't have any power storage components like batteries. I did one myself. Changed the incandscent bulb to an LED, put a bridge rectifier to turn the AC output of the generator to DC, then put a supercap in between the LED and the generator to smooth out the power flow and store some additional charge. Crude, but works. I'm trying to see what else can be done to improve it in terms of power output, light output and efficiency.