I don't know much about specifically motorsports culture, but there have been a few recent studies on cars in the Soviet Union and Eastern Bloc. If you are mainly interested in racing and not automobile culture in general, there are a couple more specific links I've given at the bottom to images/videos. The rest of what is below might or might not be of specific interest to you, but hopefully it will at least point you toward some sources that will be interesting.

Siegelbaum, Lewis H. Cars for Comrades: The Life of the Soviet Automobile. Cornell University Press, 2008.

Siegelbaum, Lewis H. ed. The Socialist Car: Automobility in the Eastern Bloc. Cornell University Press, 2011.

The former is a monograph and the second is an edited work that has a number of essays from various scholars, the full contents of which can be found here.

Cars for Comrades is much more a study of material culture - that is to say there are large parts of the study that focus on the cars themselves, which you may or may not find interesting, including the stories and details of some of the largest and most productive car manufacturing factories in the Soviet Union. Car ownership itself was, as you say, a contested thing for the Soviet government, but the actual practice of car ownership was itself, according to Siegelbaum, something that pushed people into an informal economy for car parts, repairs, etc. This mirrors many other parts of the Soviet economy, in which the favors, acquaintances, etc., were important parts of being able to secure the material goods you needed. This was only exacerbated by car ownership due to extreme shortage in the materials needed to maintain cars. Cars also acted as a kind of private space that the Soviet government had difficult infiltrating and contributed to the tension between private (or personal) and public space, especially after the Second World War.

You can also find some more material by the same author here: http://soviethistory.msu.edu/1947-2/cars-for-comrades/

The Socialist Car contains a number of essays and therefore can't really be summarized easily, but talks quite a bit more about the culture surrounding cars specifically in the eastern bloc (so not technically within the Soviet Union, but within its area of influence). None of the articles speak directly to motorsports or racing, I'm afraid, but they do speak to ways in which cars influenced the lives of people who had them.

What little I have on racing in particular:

I also dug up a couple links that I had saved when I was doing some reading on Soviet car, on Soviet sports cars in particular. I've only skimmed the material myself I have to admit, but you can find some information on this site: http://sovetcars.ru/. There are some images of cars that, to me, look like they are taken during races - but I'm afraid I don't know the details! This actually seems like a great research topic for someone going further into topics of automobile culture in the Soviet Union. Click on the American flag at the top of the page for English language if necessary.

Yet another link here on youtube shows cart racing in 1960 https://www.youtube.com/watch?v=7mkME09NHYY This was a more official kinds of racing - not an underground racing culture. This is another one in Russian language, but the visuals might be interesting to you. The narrator is giving a generic kind of account of what is going on as you might imagine from this sort of video (hopefully you'll know what I mean when you see the thing).

I don't know about modifying cars, but there was a prominent self-made car scene. Here are some photos of survivors, with captions (in Russian, but Google Translate ought to help out).