Assuming nothing as simple as the device being partially choked by scale or water-treatment additive. Also assuming the water-inlet strainer is not choked enough to starve the injector under full draw:

My first thought is the injector is taking in air through that leaky water-valve. The valve will emit water when the injector is off, but being on the suction side of the device may allow enough air in to stop the injector's action.

Ditto anything else on the water line. You don't say what this injector is on, though seems directly on the water-tank. It's not easy to determine from the photo, so I took the liberty of copying it to be able to enlarge and rotate it to view from different angles; a bit puzzled by what's "below" it in your original.

If as seems the installation is directly on a tank wall, it ought be almost as reliable as if actually inside the tank.*

Next, try the condition of the overflow valve in the injector itself. It is in the suction half of the injector, depending on the depression in the combining-cone to close it.

'

Repairing the valve may be possible depending on its design. It looks like a plug-cock. Leaking past even new O-rings suggests a worn spindle. Refurbishing the plug and bore may be as straightforward as lapping them together.

Is the plug retained by a nut and washer on a threaded tail? Also a potential leak-path though not so likely because the main sealing surfaces are the plug and taper-bore walls.

'

Finally, the control-rod connection looks quite rigid. It may be better if slightly flexible so any slight misalignment does not put unfair radial strains on the plug and seals.

'

'

*I asked because if on a loco or traction-engine with a water-tank in a tender or driving-truck, I'd suspect also the flexible hose needed in the water line. As I think DAG Brown points out in his book on injectors, the simple hose pushed onto the pipe-end can allow very fine air-leaks not obvious if the valve is upstream of the hose. One cause is the hose material stretching and age-hardening slightly.