Does weight influence Earth's spin?
The Earth does spin like an unbalanced top. The Earth's rotation axis is not fixed. It instead moves in a complex manner due to a combination of external torques exerted by the Moon and Sun, a torque-free nutation due to the oblate shape of the Earth, and also due to changes on and in the Earth.
The torque-induced motions are called precession and nutation, distinguished by period. The largest and slowest of these motions is the axial precession. This causes the Earth's rotation axis to trace out a cone over the course of 26000 years.

(source: nasa.gov)
The torque-induced nutations are also cyclical motions induced by the Moon and the Sun. These are much smaller in magnitude and have a much shorter period. The largest of these has a magnitude of about 20 arc seconds and a period of 18.6 years. All other nutation terms have much smaller magnitude and have shorter period.
The torque-free nutation would have a period of about 305 days if the Earth was solid. The oceans, the atmosphere, and the outer core alter this. The Chandler wobble has a period of about 433 days and a magnitude of less than an arc second. Because the Chandler wobble isn't as predictable as are precession and nutation, it's lumped into a catch-all category called &quot;polar motion.&quot; The redistribution of water over the course of a year (e.g., snow on Siberia in the winter but not in the summer) results in a yearly component of the polar motion.
There are lots and lots of other factors, all small. Polar motion is observed after the fact.