Was censorship something that only happened post hoc, or could a writer, or a filmaker, etc. contact the relevant authorities before publication?
The censorship in the Soviet Union worked, for the large part, on the imprimatur principle, meaning that the author had to receive the permission of the censorship to actually publish more or less anything. This was aided by the fact that virtually all establishments capable of publish any book was collectively owned, what essentially meant it was owned and governed by the state.
On the 6th June 1922, Soviet National Committee proclaimed the General Office of Literature and Publications (commonly known under the abbreviation 'Glavlit' derived from Главное управление по делам литературы и издательств). This institution was responsible for controlling all the published materials, including books (literary works and documentaries), newspapers and journals, and since 9th February 1923, also scripts of movies and theatrical performances (this was covered by the General Committee of Theatrical Repertoires, or Glavrepertkom, a subdivision within the structures of Glavlit). In theory, Glavlit was subject to the equivalent of the cabinet (SNC, then Council of Ministers of the USSR) but in reality it was almost completely controlled by the Central Committee of the CPSU. The censorship of the radio broadcasts was among the duties of All-Soviet Committee of Radio Broadcasts, created in 1931 and then divided into the General Office for Radio Information and General Offices for Radio Broadcasts within the structures of the Ministry of Culture. On 16th May 1957, the former institution was renamed to USSR State Committee for Television and Radio Broadcasting (Гостелерадио СССР, from Государственный комитет СССР по телевидению и радиовещанию), covering all types of broadcasts, including the new television technology.
As a general rule, no printing shop, cinematographic studio or a theater would have allowed any book to be printed or a spectacle to be produced (or at least, presented to an audience) without the the proper permission of the Glavlit (the same applied to other institutions mentioned above in case of broadcasts). This meant that to have the chances of publication, the author had to submit the work to the authorities, where it has been read more or less thoroughly and then, depending on their personal opinions and disposition towards the author, censors were either removing the 'unacceptable' parts or were discussing how they can be changed to comply with the requirements. So, technically, it didn't mean that getting an censor's approval was difficult by definition - although any form of direct or indirect critique of the government or the general party line, mentions of concentration camps or terror practices or description of 'debauched' practices were strictly off-limits, a lot of writers, especially those specializing in children or popular literature either did not have troubles with censorship or knew well what to expect and simply avoided any potentially problematic subjects. Conversely, authors highly critical of the government, incumbent or historical who tried to include their opinions in their books had a lot of trouble to get through censorship.
This did not mean that all of the publications were censored, as industrious authors were able either to procure publications by their own means (so-called 'samizdat' or self-made publication, from Russian самодельные издательство). Due to the state control over the market and resulting troubles related to the procurement of the larger quantities of paper, not to mention the printing machines, samizdat was usually published using typewriters and carbon paper, photography, duplicating devices and, in rare cases, small printing machines, what severely limited the number of copies, not unlike in the case of the underground zines in the West. Some authors were also using the method commonly called 'throwing a bone', i.e. including some elements blatantly contradicting censorship rules (and never intended to be a part of the final work) and hoping that between the comparatively less outrageous remainder of the work and more or less lackluster diligence of censors, they will be asked to remove only the inadmissible parts, leaving the actual text unscathed.