In particular, was there any move by anarchist in Hungary to regect all the weak, unsuccessful and narrowly supported governments that tried to take control of the country, and create self-governing communities according to anarchist principles, like in parts of the former Russian Empire, and what was destined to happen in Spain in 1936?
There were and depending on the country you're referring to had different make-up and alliances with socialist parties and sects.
From the initial outbreak of the war nations across Europe immediately descended on the anarchist and socialist elements within their countries to prevent any possibility of building support these organizations had. For decades the anarchist movement in its literature suggested seizing opportune times to create revolutionary movements and nation-states immediately tied the assassination of the Archduke Franz Ferdinand to the standing anarchist practice of "propaganda by the deed." By the beginning of World War 1, anarchists had assassinated a number of heads of state, from President William McKinley in 1901, the Prime Minister Jose Canalejas of Spain in 1912, King George I of Greece in 1913, and dozens of bombings and other violent acts.
But while the governments of Europe had focus on the anarchist movement, the anarchist movement itself was not entirely certain of how to handle the war itself. The Russian Peter Kropotkin, one of the anarchists arrested in the run-up to the first World War, penned a letter in the anarchist newspaper Freedom suggesting anarchists support the war against Germany for its history in suppressing anarchist and socialist movements. The letter was supported by a number of anarchists including one of the founders of the Anarchist Black Cross, and Jean Grave editor of the anarchist-communist periodical Le Revolte. Following the publication of the letter, the anarchist movement held a series of debates on the focus of anarchists, which held responses such as Errico Malatesta titled "Anarchists have Forgotten their Principles." The debates continued through the wars end but coming to a peak in 1916 with the release of the "Manifesto of the Sixteen." The release of the manifesto was described by the anarchist Paul Avrich in his book The Russian Anarchists, "The issue of the war effort caused an almost fatal split in the anarchist camp..."
At the same time of the crisis of which side of the war to support (or based on your reading which side to not fight against as much), the anarchist movement had already began to fracture into a number of sections. Following the Anarchist International meeting in 1907, the movement was in contention over various issues namely on the stance of anarchist unions and general trade unionism, made most evident by its debates between the French syndicalist Pierre Monatte and the Italian anarchist-communist Errico Malatesta. In its conclusions there was little concrete results, as described by the anarchist historian Alexandre Skirda,
[T]he concrete results of it [the Conference] were limited, the links forged in this way rather loose, and no real decision leading to concerted practical undertakings was arrived at. Which explains why there were no other congresses for many a long year: the organizations and most of the individual militants were absorbed by their national and day-to-day tasks... in the years leading up to 1914, when international connections were so sorely needed if the looming threat of war was to be defused.
In the face of this disorganization, anarchists in various countries saw the looming Russian Revolution and the growing workers movement following the initial strikes of March of 1917. Many anarchists did not know whether or not to join the various socialist parties and whether the long held distances between the socialists and anarchists would come to an end.
Beginning in Hungary, the anarchists had been intimately tied to the other socialist-left. One such group, the Union of Revolutionary Socialists who had been apart of the 1907 Anarchist International, then later split into various socialist and Bolshevik supporters, most of whom joining the Hungarian Communist Party by 1916. As the war continued the socialists pushed for war-resistance and mutiny, which began to occur en-masse in 1918 beginning with the Cattaro mutiny and the minor rebellion in Pecs. The various smaller rebellions began to spiral, in large part of influence by the rising Russian revolution, with a number of socialists and worker councils formed, culminating in the declaration of the short lived Hungarian Soviet Republic in 1919. As the Soviet republic was declared, just as they had been in Russia, anarchists who were once supportive of the Soviet state became disillusioned, and a number of Anarchists who had once joined the Communist Party of Hungary left the party and attempted to form its own group the Anarchist Union, which quickly fell apart as they were arrested by either the Soviet Republic or the Romanian army who entered Budapest in early August.
Similar short lived revolutions by alliances between the socialist parties and the anarchists happened across central Europe. In Greece anarchists organized and took part in many labor actions, most notably the anarcho-syndicalist miners strike in Serifos. The strike was immediately brought down upon by Greek authorities, who quickly arrested the organizing committee. In Serbia the anarcho-syndicalist group Direktasi had existed as a faction of the Social Democratic Party, which were expelled in 1909 and 1910, after which members were arrested, forced underground, left to France and other Serbian immigrant communities, or joined the socialist and communist parties (League of Communists of Yugoslavia as one example).
The largest of these movements and similar in composition to the Hungarian Soviet Republic occurred in 1918 to 1919 following the defeat of the German Empire, which dubbed itself the Bavarian Socialist Republic. Anarchists and socialists were in alliance and when the "republic" was on-going anarchists were named to several positions within the organization. The republic attempted to abolish money, radically reform education, create workers councils and communist workers groups. The republic was rather short lived, by May 1st, "Regular Army troops [Freikorps] volunteers entered Munich and overthrew the Communist regime, massacring several hundred persons, including many non-Communists, in revenge for the shooting of a dozen hostages by the Soviet." (from Rise and Fall of the Third Reich by William Shirer)