Why is the slowest point of Earth's rotation in the middle of the year? The following image is taken from Wikipedia's article on the leap second.

Why is the slowest point each year in the middle of the year around July? Does being further from the Sun cause the Earth's rotation to slow down? What's the mechanism in play here?
 A: I don't have access to the paper but the abstract from Carter(1984) suggests that the major cause of rotational variations is due to "the exchange of angular momentum between the atmosphere and the mantle."
This is echoed by Earth Rotation Variations from Hours to Centuries:

Variations with periods of five years or less are driven primarily by exchanges of angular
momentum with the atmosphere

One could hypothesize that the middle of the year corresponds with some  weather pattern that maximizes atmospheric momentum.
A: I suspect that cycle is connected to the variation in the orbit of the Earth and Moon around the Earth-Moon barycentre (EMB).
From the IERS

Measuring the irregularities of the Earth's rotation
The variability of the earth-rotation vector relative to the body of the planet or in inertial space is caused by the gravitational torque exerted by the Moon, Sun and planets, displacements of matter in different parts of the planet and other excitation mechanisms. The observed oscillations can be interpreted in terms of mantle elasticity, earth flattening, structure and properties of the core-mantle boundary, rheology of the core, underground water, oceanic variability, and atmospheric variability on time scales of weather or climate. The understanding of the coupling between the various layers of our planet is also a key aspect of this research.

As I mention in this Astronomy.SE answer, the size of the Moon's orbit varies over the course of the year. The maximum orbital radius occurs when the Earth-Moon system is near perihelion or aphelion. That could explain the dip near aphelion in the UT1-UTC graph, but we'd also expect to see a dip near perihelion. My guess is that there is a corresponding dip, but it's less noticeable because it has a shorter duration, since the orbital speed of the EMB is at its maximum at perihelion. However, I may be wrong. I get dizzy trying to keep track of all these rotations and revolutions. :)
Here's a daily plot (courtesy of JPL Horizons) of the Earth-Moon distance for 13 (anomalistic) lunar months.

Here's the corresponding plot of the distance from the centre of the Earth to the EMB.

The mean distance over that period is ~$4677.675$ km.
That Astronomy.SE answer has a link to a live script you can use to make your own plots. I have some info about the impact of the Earth-Moon orbit on the time of perihelion and aphelion here.
