Forget that idea! IT IS EXTREMELY BAD.
Any extra power applied in that manner, other than direct hydraulic power (obtained by getting either more water head or water flow on/to the turbine), comes at a very, very, very large cost.
If you add say a 500 W pump, due to pump efficiency of, say 0.54 for a low power pump, the pump will only give an output of 270 Watts.
That 270 Watts New output power will be applied to the turbine. Due to turbine efficiency, say 0.70, only 270*.70 = 189 Watts will be recovered as output from the turbine.
True you got 189 New Watts output, but it cost you 500 Watts input.
The Efficiency of that transaction was 189/500 = 37.8 %
The efficiencies combine by the multiple of each power source.
0.700 * 0.540 = 0.378
It would be much better just doing this by adding more water flow (or more head) directly to the turbine, without the pump. Adding the same 500 W in direct hydropower on the turbine would get you 500 * 0.7 = 350 W instead of only the 189.
To make that attractive, you would have to buy and install the pump and its power source at a cost cheaper than adding the direct hydropower in either flow, or head, or a combination of both. And you would still have to pay for running that pump power source generator, a gasoline generator, or a solar generator, or grid power.
Let's say you go solar for that. You need 500 Watts of power supplied to the pump. For normal Si solar panels, efficiency say 17%, you have to install 500/0.17 = 2941 watts, say 3000 watts. Now with 3000 installed and only the same 189 W output from the pump, that's 189/3000 = 0.063 = 6.3 % efficiency on that little addition. And that probably cost you 3000 x $4/watt installed cost = 12,000. That's a bunch of money for only 189 watts total output.
If you just used the 500 Watts coming off the solar panels, you would get 500 watts, not just the measely 189 watts from the turbine.
Assuming no batteries and inverters for the solar panels. Add that for night work and/or an AC pump. No electric line losses counted either.
Furthermore, turbines are built for a specific water flow and head. If you change either, the turbine efficiency will be less than 0.70
Either go full hydro, full solar, some independent combination of both, but NEVER, NEVER, NEVER feed one to another and expect to gain much. The penalty of hooking them together is Eff 1 x Eff 2 x Eff 3 and so on.
True you can do it, but as you can see, you play a fools game.
"People will work for you with blood and sweat and tears if they work for what they believe in......" - Simon Sinek