Short answer: It might be impossible "long term". It's problematic because of the size of the Moon and the proximity to the Sun.
The Earth-Moon system in empty space could hold several more moons (like the Pluto-Charon system). The problem is proximity to the sun in addition to the size of the Moon. Planets closer to their star have a harder time holding onto moons in part cause the Hill sphere is smaller, in part cause the tidal forces are that much bigger (and those 2 factors may go hand in hand, not in addition to). But it's not just coincidence that Mercury and Venus don't have moons, they're also less likely to hold onto Moons.
Mars has a smaller Hill Sphere than Earth due to it's lower mass but it's 2 moons are very tiny. Also, one of Mars 2 moons might not be there that much longer. It may crash into Mars in as little as 100 million years.
Earth's proximity to the Sun gives the Earth a Hill Sphere of about 1.5 million km, and the true region of stability (long term) is about 1/3 to 1/2 of that, so about 500-750,000 km which isn't all that much further out than the Moon is currently.
Our Moon's average distance is 384,000 km, but our moon has a slightly eccentric orbit so it's distance varies between about 363,100 and 405,700 km. Source.
The 2nd factor is the relative mass between the Moon and the planet. Mars has 2 moons but it's moons are very small and they have very little gravitational effect on each other. Our Moon is 1/81st the mass of the Earth which is the largest Moon to planet ratio in our solar-system.
Using the simplified hill sphere formula, the cube root of (the ratio of the masses/3). Basically the cube root of (1/243) gives Earth's moon's a hill sphere of about 16% it's distance to the Earth. Any Earth orbiting object that passes within the Moon's hill sphere is obviously not going to be in a stable orbit. In fact the region of instability probably extends well past that 16%. How far past, I'm not sure, but I can safely say that anything even remotely close to the Moon's orbit couldn't orbit the Earth for any period of time.
Using 16%, the clearly unstable region is between 300,000 km and 470,000 KM. It would be impossible for an object to orbit the Earth at a greater distance than the Moon. Solar and Lunar perturbations would be too big.
If it's possible for Earth to have a 2nd moon at all, it would need to orbit quite close to the Earth. Probably well inside 300,000 KM. The closer to the Earth the more the Earth's gravitation would be dominant, so if I was to guess, I think the Moon would need to be quite close, maybe in the 50,000 km range - but that's just a guess.
For comparison, Jupiter's 3 moons in orbital resonance; Ganymede, the largest of the 3 has a hill sphere is about 3% it's distance from Jupiter. The 1:2:4 orbital resonance works out to the 1.5th root of 2 or about a 1.59 to 1 orbital distance ratio. Those 3 moons also have very nearly circular orbits. If the Moon's orbit was nearly circular, there might be a stable resonance 1.59 times closer to the Earth, but because of the Moon's orbit is measurably eccentric, I don't believe the 2:1 orbital period ratio would be long term stable because there's too many wobbles in it. I think your best bet for a stable orbit for a 2nd moon would be very close to the Earth, probably in the 50-100,000 KM range (but that's a rough guess).
It's worth noting that "long term stable" isn't a precise term cause there's no clear cut-off. A thousand orbits? A million? A billion? Lifespan of the Sun?
I realize that "maybe, maybe not" isn't an answer, but I wanted to touch on some of why it's difficult. A captured satellite couldn't get captured into a relatively circular near earth orbit because the incoming velocity is too great. In theory, if there was enough debris, a satellite could form close to the Earth and perhaps be stable for relatively long term, but the Moon would to some extent prevent such a formation in a similar way that Jupiter prevents the asteroid belt from coalescing into a tiny planet. (the low mass of the asteroid belt plays a role in that too, perhaps a bigger role), but Jupiter is a factor. Article here