Does an increase in inlet temperature affect fuel flow and power output?
Most civil aircraft engines are "flat rated". See What is a flat rated engine?



That means, the thrust (power) produced is constant (flat), as ambient temperature increase, up to a certain temperature, and then the turbine inlet temperature limit is reached, and fuel flow and thrust is reduced as the ambient temperature continues to increase. 

So, thrust is initially constant as inlet temperature increases, then decreases. It's a bit harder to predict what fuel flow will need to do to achieve this behaviour. It will certainly reduce in the temperature range where thrust is reducing, but in the cooler temperatures before that, it is probably increasing to keep thrust constant. As the chart shows, in this region EGT and N1 rpm are increasing. I expect fuel flow is probably showing the same behaviour as N1.

Thrust cannot actually be measured in flight, so most engines use either N1 rpm, or engine pressure ratio, (turbine exit perssure/engine inlet pressure) as a way to predict thrust. GE typically uses N1, whereas Pratt &amp; Whitney and Rolls Royce usually use EPR. Using EPR is technically better, beacuse it will account for engine deterioration, where as using N1 assumes the fan mass flow remains the same for a given rpm as the engine degrades. But EPR requires 2 pressure measurements, which are less reliable, so most engines that use EPR have N1 control as a backup mode.  See 
What is the difference between EPR and rotor speed as thrust setting parameter?