Fuel pressure?

First the pump running for just a couple of seconds is proper operation. Fuel pressure should hold for weeks, or at least multiple days, but in case it does bleed down a little, that two-second burst is designed to get pressure back up for starting. After that, the pump gets turned back on during engine rotation, (cranking or running). Safety is the reason the circuit works this way. If a fuel line is ruptured in a crash, raw fuel would get dumped onto the ground and be a serious fire hazard. Instead, with a ruptured line, there can't be any fuel pressure. With no pressure, no fuel will spray from the injectors. The engine will stall, then, with no signal pulses from the crankshaft position sensor or the camshaft position sensor, the Engine Computer turns the fuel pump off again. That stops it from pouring gas on the ground.

Cycling the ignition switch multiple times is a test that can prove fuel pressure is dropping down too quickly. Pressure can be lost due to a leaking injector, a leaking check valve in the fuel pump, or, depending on the type of fuel supply system, a leaking fuel pressure regulator.

It's easiest to watch the pressure drop over time with a fuel pressure gauge, but you found a clue to do the same thing. That's the hard starting / sputtering. If you have a hose pinch-off pliers, use it to block the fuel return hose if your vehicle has one. That will be the smaller of the two connected to the fuel rail on the engine. If pressure still fades away, block the larger fuel supply hose and watch again.

Most commonly the pressure will continue dropping with both tests. That leaves an injector which is by far, the most common cause. On some engines, you can remove the spark plugs and find one that's wet, or you can see fuel dripping into the cylinder, but that's not conclusive. Except for the newer direct injection models, gas leaks from an injector into the intake manifold, not directly into the cylinder where you can see it. Then we have to remove all of the injectors still connected to the fuel rail. The leaking one will be evident by the wetness on its tip.