According to Wikipedia the entire Yak-9U was covered with Bakelite (not sure if it was just that variant, the page isn't clear). What was the reason for this and was Bakelite used as a coating on any other aircraft?
The aircraft used bakelite impregnated wooded stressed skin (basically molded Bakelite birch plywood). For example, airvectors notes:
In the late spring of 1942, the increased availability of aviation aluminum alloys led to the development of a reconnaissance variant of the Yak-7 with a new wing, featuring metal H-section spars with Bakelite-impregnated wood skinning.
This development variant became the Yak- 9. Yakolev had used this method construction earlier in Yak- 2. This method of construction was used in a number of other aircraft too. For example, the Mikoyan-Gurevich MiG-3 used a combination of Aluminum and bakelite infused plywood:
... the fighter was of mixed construction: the forward and central sections were of welded steel tubes construction covered with Dural panels, the rear- and tail- sections were semi-monocoque structures with a skin of bakelite-impregnated plywood over wooden longerons and stringers.
The wing was a stressed-skin structure: the flat center section, which was tapered in thickness and chord, had a flush-riveted Dural skin over a steel and Dural structure, while the dihedraled outer wing panels, which were also tapered in thickness and chord, possessed a skinning of bakelite-impregnated plywood over a wood structure.
These are early composites which increased strength and water resistance. This method was especially attractive to the Soviets due to the early loss of industry (Aluminum plants) and wartime scarcities.
This method of construction was used in a number of other aircraft with various modifications, like the Fokker DXXI among others. This method was used, albeit on a smaller scale in the US too (There was little incentive there to use wood for aircraft construction).
The Bakelite finish was for the wooden parts/variants.
Bakelite back then was a novelty, its usage in airplanes strengthened the wood, made it moisture resistant, and offered a smooth aerodynamic light-weight finish.
It was also highly flammable:
The LaGG-1 was built with highly flammable delta-drevesina (delta wood), plywood sheathing, and bakelite ply, and burned so easily that the pilots sarcastically joked that LaGG in Russian stood for Lakirovannyi Garantirovannyi Grob (varnished guaranteed coffin).
The LaGG-3 article on Wikipedia offers more details on this method of construction:
[The] airframe was almost completely made of timber, with the so-called delta-lumber (a wood-plastic composite composed of very thin, 0.35 to 0.55 mm, wood veneer and phenol formaldehyde resin [Bakelite], baked at high temperature and pressure) used for the crucial parts. This novel construction material had tensile strength comparable to that of non-hardened aluminum alloys.
Sources: MOISTURE RESISTANT FINISHES FOR AIRPLANE WOODS (NACA, 1920), and Russian Aviation and Air Power in the Twentieth Century By John Greenwood.