If the universe is sufficiently large enough, there should be a black hole in every direction of the sky. So, could there be a game of tug-of-war going on between all of the black holes in the universe? It could explain the accelerating rate of expansion due to there being more black holes that have grown larger. The universe wouldn't actually be growing, it would be stretching.
Answer
Outside of its event horizon the gravity from a black hole is exactly the same as the gravity from any object that isn't a black hole. So black holes just make up part of the overall stuff in the universe. There are no effects on the universe expansion that are a special result of black holes being present.
The gravity of the black holes certainly does affect the expansion, but black holes affect the expansion in the same way as all the other matter. The geometry of the universe is approximately described by the FLRW metric, and the effect of the matter (black holes and all) is neatly summarised by the Friedmann equations. If we ignore dark energy (and assume pressureless matter) then the acceleration of the expansion is given by the second Friedmann equation:
$$ \frac{\ddot{a}}{a} = -\frac{4\pi G}{3}\rho $$
where $\rho$ is the average density of all the matter including the black holes. And since the density $\rho$ is positive that means $\ddot{a}\lt 0$ i.e. the expansion must be decelerating. Black holes cause the expansion to decelerate just like all the other forms of matter. The only way to get an acceleration is to include the dark energy $\Lambda$ in which case our equation becomes:
$$ \frac{\ddot{a}}{a} = -\frac{4\pi G}{3}\rho + \frac{\Lambda c^2}{3} $$
and we get an acceleration when:
$$ \frac{4\pi G}{3}\rho < \frac{\Lambda c^2}{3} $$
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