Okay, this may be two weeks after you asked but I hope you are still interested, I will only answer the first question (since that alone already takes some time), namely if "Nestor Makhno and his Black Army really saved the Bolsheviks in the nick of time".

Here is a short expanation of what exactly was happening at the time we are talking about:

In September 1919, the Civil War in Ukraine is still raging on. The White Army of General Denikin has major successes as it captures large areas of Ukraine and southern Russia including (among others) such cities as Tsaritsyn (the later Stalingrad), Kyiv, Kursk and Odessa. The Red Army is in disarray and the route to Tula, the center of the armament industry, is open, capturing it could be a major achievement for the Whites. Denikin is optimistic, he even expects to reach Moscow and crush the Bolsheviks before the end of the year.

The Makhnovists, on the other hand, had been in retreat for some time until they arrived at the area around Uman (hundreds of kilometers from their home area around their "capital" Guliaipole). They were hampered by thousands of wounded and sick soldiers that travelled with them, lack of ammunition and in danger of getting encircled by the White forces.

However, on September 25th the Makhnovists started a counterattack led by Makhno himself exploiting a gap in the Whites frontline, the Whites poor coordination aswell as them underestimating the peasent armies.

This "Battle of Peregonovka" was a full success for the Makhnovists and it was a deciding one since it had opened the Whites rear to the Makhnovists. Denikin, in his focus on Moscow, had overstretched the frontline and underestimated the danger of the Makhnovists to exploit this. This now enabled the Makhnovists to perform an impressive charge through the Ukraine towards the Dnepr (and Guliaipole), gaining control of most of southern Ukraine in only a few days and crushing the White troops and garrisons, that lacked coordination and information to respond to that sudden push, on their way.

The biggest problem resulting from that for the Whites was the Makhnovists cutting their supply lines by taking over or destroying the storages, armouries and railways. The Whites also had to redirect significant amounts of troops that were previously supposed to contribute to the march on Moscow since the Makhnovists even came close to threatening Denikins headquarters. At he same time large insurrection spread behind the Whites frontline.

As to the question if all of this "saved the bolsheviks" or even "caused the defeat of the White Army", it is of course a bit more controversial and debatable, but it seems that a large amount of historians support the notion that the Battle of Peregonovka and its consequences played an important role in the Civil War.

Footman noted that "There is some justification for the claim that Peregonovka was one of the decisive battles of the Civil War in the south" while Palij declared that "The developments resulting from [the] defeat [at Peregonovka] decided Denikin’s fate" and Skirda went as far as to say that "Makhno had broken the back of the great Denikinist offensive that the Red Army had failed to halt [... .], the battle of Peregonovka had been the crucial feat of arms in the civil war".

Colin Darch explains how Makhno had "[earned his place] in the broader historiography of the revolution and civil war" and cites some other historians to show how widespread the notion of the idea of the Makhnovists deciding the war was:

"Malet quotes from an obituary of Makhno asserting that ‘it is certain that Denikin’s defeat owed more to the peasant insurrection under the black Makhnovist banner than to the successes of Trotski’s regular army’. [...] Chamberlin argues that [... .] Makhno was able ‘to play a most devastating role’ behind Denikin’s lines. Mawdsley cites White officers to the effect that the removal of regiments from the frontline to deal with Makhno allowed the Red Army to turn the White flank at Orel’ and launch a counter-attack. Lincoln agrees, pointing to the destruction of the armoury at Berdiansk by Makhno in early October as a heavy blow, denying the Whites the artillery shells needed for the assault on Orel’. Pipes says that the diversion of White regiments to fight Makhno ‘had a very detrimental effect on the battle for Orel and Kursk, which decided the Civil War’."

He mentions that "A few voices disagree" pointing to two people: "Figes, quoting Trotsky, argues that ‘the entire fate of the Soviet regime hinged on the defence of Tula’ where Russia’s main armaments factory was located. Litvinov believes that even in the absence of a breakthrough at Uman’, the Bolsheviks ‘would have managed to organise the defence of Tula and Moscow"

Darch himself argued that "Makhno’s victory at Peregonovka was a contributory factor to Denikin’s defeat, but not necessarily the decisive one", he notes the insurrection behind Denikins lines which "[Makhnos] return to the left-bank sparked" but which was actually prepared by "Denikin’s policies, driven by his narrow class interests" as a major factor for his loss.

Overall it seems that the Makhnovists certainly played an important role in the Civil War, tho how important exactly remains debated among historians.

Literatur used:

Colin Darch (2020). Nestor Makhno and Rural Anarchism in Ukraine, 1917–21.

Michael Palij (1976). The Anarchism of Nestor Makhno, 1918–1921. An Aspect of the Ukrainian Revolution.

Alexandre Skirda (2004). Nestor Makhno: Anarchy’s Cossack. The Struggle for Free Soviets in Ukraine 1917–1921.

David Footman (1961). Civil War in Russia.

Аleksandr Shubin (2010). "The Makhnovist Movement and the National Question in the Ukraine, 1917–1921" In Anarchism and Syndicalism in the Colonial and Postcolonial World, 1870–1940. The Praxis of National Liberation, Internationalism, and Social Revolution by Steven Hirsch & Lucien van der Walt (Edit.).