Do variations of the Earth's magnetic field affect gravitational acceleration at Earth's surface, and if yes, by how much?

The effect of variations of Earth's magnetic field on gravitational acceleration is completely negligible, as the following rough estimate shows.

The first question is what to take as the magnetic field inside the Earth. Outside, a dipole field is a good approximation, but if we continue to use that field inside, the field strength gets stronger and stronger as we go toward the center. The field strength goes as $1/r^3$ and the energy density as this square of this, so $1/r^6$. Integrating this over the interior of the Earth gives a divergent integral for the energy in the field.

But the Earth's magnetic field is not really the field of a point dipole at the center of the Earth. Instead it's the field of a messy, rotating, convecting, liquid dynamo the size of the Earth's core. Pictures of it usually make the field lines look like spaghetti. So let's not try to actually model the field inside the Earth. Let's take a very crude estimate.

This article says that the field strength averaged over the outer core is 25 Gauss (0.0025 Teslas), or about 50 times stronger than the field at the surface. Let's just assume that this is the average throughout the interior of the Earth. Maybe the field is stronger in the inner core, but it's definitely weaker outside the outer core, so we're probably doing an overestimate of the effect. But it's going to turn out tiny anyway.

Using this average field strength, the magnetic energy density $B^2/2\mu_0$ avereages 2.5 Joules per cubic meter.

The radius of the Earth is $6.4\times10^6$ meters, so the volume is $1.1\times10^{21}$ cubic meters.

Multiplying the energy density by the volume gives a magnetic energy inside the Earth of $2.7\times10^{21}$ Joules.

Dividing by the speed of light squared gives an equivalent mass of 30,000 kilograms.

The mass of the Earth is $6.0\times10^{24}$ kilograms, so the mass-equivalent of the magnetic energy comprises only a fraction $5.0\times10^{-21}$ or 0.000000000000000000005 of the total mass. The gravitational acceleration $g$ is proportional to the mass of the Earth, so the same negligible fraction of $g$ is due to magnetic energy. This fraction is not measurable by any current technology.

Even if the magnetic field completely disappeared, it would not affect gravitational acceleration.

The earth's field is very weak as these things go. Its energy density isn't big enough to produce a measurable change in the earth's gravity.