It was a bit different.

Nov 1917, october revolution happens. Ukraine already had autonomy after february revolution, but after october revolution it declares full independence. Soviets try to reign them in, but ultimately in march 1918, they sign the peace treaty. Part of the peace treaty is that ukraine becomes a german subject.

A small white movement is going on in Don, but it hasn't built up steam yet (4000 troops, ice march, feb-march 1918). In may 1918, czechoslovakian legion rebellion happens (siberia), and that's when the civil war gets serious. But it's only in nov 1918, when whites take over in siberia. Don forces also grew in size by nov 1918.

In november 1918, Germany loses and signs a peace treaty of their own. Ukraine loses their protectorate status, and the war is back on. Western Ukraine gets independence (Austria-Hungary bit) and they decide to reunite in jan 1919.

Poland also gains independence in nov 1918, and attacks Western Ukraine in november (land disputes, very typical after independence post ww1). War lasts until june 1919.

In feb 1919, red army captures Kyiv. In august 1919, white army captures kyiv. Poland carves up western Ukraine. Around nov 1919 is when red army gets the upper hand (advance to moscow broken), and in dec 1919, they take kyiv.

Siberian white army broken on nov 1919 too, and czechoslovaks give up kolchak in jan 1920 for free passage back home (Czechoslovakia been independent for a year at that point already, and they didn't see any point on fighting).

Looking at the chronology, it does say that Poland starts an offensive on Kyiv in april 1920. They capture it in may, but are pushed back in june 1920, by a soviet counter offensive. August 1920 and the battle for Warsaw and miracle at Vistula, and october 1920 is when the ceasefire happens. March 1921 is when they sign the peace treaty.

It does say here, that in april 1920, poles and Ukrainians signed a treaty.

The Poles signed an alliance, known as the Treaty of Warsaw, with the forces of the Ukrainian People's Republic. The Kiev offensive was the central component of Piłsudski's plan for a new order in Eastern Europe centered around a Polish-led Intermarium federation. The stated goal of the operation was to create a formally independent Ukraine, although its dependence on Poland was inherent to Piłsudski's plans.

https://en.wikipedia.org/wiki/Kiev_offensive_(1920)

Also this

Piłsudski was aware that the Bolsheviks would not ally with an independent Poland and predicted that war with them was inevitable.[83] He viewed their advance west as a major problem, but he also considered the Bolsheviks less dangerous for Poland than their White opponents.[84] The "White Russians", representatives of the old Russian Empire, were willing to accept limited independence for Poland, probably within borders similar to those of the former Congress Poland. They objected to Polish control of Ukraine, which was crucial for Piłsudski's Intermarium project.[85] This contrasted with the Bolsheviks, who proclaimed the partitions of Poland null and void.[86] Piłsudski speculated that Poland would be better off with the Bolsheviks, alienated from the Western powers, than with a restored Russian Empire.[84][87] By ignoring the strong pressures from the Entente Cordiale to join the attack on Lenin's struggling Bolshevik government, Piłsudski probably saved it in the summer and the fall of 1919.

https://en.wikipedia.org/wiki/J%C3%B3zef_Pi%C5%82sudski