Would injecting water into a jet engine’s exhaust increase velocity?
Spraying water into the exhaust stream will cool down the exhaust flow. The energy needed to heat and evaporate the water needs to come from somewhere, after all. The cooler and denser exhaust flow will slow down, so its impulse will become lower, reducing thrust. While the mass flow will go up, its speed will be reduced. Since water enters the flow as a liquid and leaves the nozzle as a gas, its evaporation enthalpy is irretrievably lost and removes energy from the exhaust stream. The slightly higher mass flow in combination with a lower speed will result in a lower impulse.

If you spray water into the intake flow (or after the first few compressor stages), the same evaporation enthalpy will allow a higher mass flow in the intake, so a higher mass flow in the combustion chamber can be achieved. Also, the cooling of the intake air allows to add more fuel in the combustion process, so more thrust can be created. Of course, this is bought with higher fuel flow.