The Italian prime minister, Giuseppe Conte, has appealed for unity within his coalition government after a tumultuous few weeks in which three senators from the Five Star Movement (M5S) absconded to the far-right League party and the education minister quit.

Lorenzo Fioramonti resigned last week after the government approved a budget for 2020 that did not contain an increase in spending on education. Fioramonti, who in November declared that Italy would be the first country to teach climate change in schools, has hinted at forming a parliamentary group that would include other disgruntled M5S politicians.

In what was hailed as a victory against the far right, M5S in September went into coalition with the centre-left Democratic party (PD) after the collapse of its alliance with Matteo Salvini’s League when he brought down the government in a move to force elections. But the past four months have been marred by constant bickering and scant action on policy.

In a speech on Saturday, Conte said he was dividing the education ministry in two, naming Lucia Azzolina, also from M5S, as minister for schools, and Gaetano Manfredi, a non-party technocrat and rector of Naples University, as minister for universities and research.

Conte also took a swipe at Salvini, who stills looms large over Italian politics, calling his behaviour “insidious”. As the he outlined a programme that included reforms of the justice system and state bureaucracy, he added: “Politics doesn’t need conflicts. Polemics and marking out our differences do not help.”

The differences between M5S and PD have been laid bare since the government came together and few expect it to last. The alliance has failed to devise any strong policies or respond effectively to industrial crises, and has backtracked on key initiatives, such as a tax on plastic. Even though the budget passed without controversy, the financial strategy contains nothing that will tackle high unemployment or revive Italy’s long-stagnant economy.

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“This government is going nowhere,” said Wolfango Piccoli, the co-president of the London-based research company Teneo Holdings. “It is spending most of its time arguing and not much gets done at all. This is unlikely to change in the foreseeable future.”

The main thing keeping the alliance together is the perennial threat of Salvini. Polling at approximately 32%, the League remains the biggest party in Italy, while support for its far-right ally, Brothers of Italy, has risen in recent months. In coalition with Brothers of Italy and Silvio Berlusconi’s Forza Italia, the League seized power in the former leftwing stronghold of Umbria in late October.

The close-knit alliance is campaigning vigorously ahead of the next crucial test in Emilia-Romagna on 26 January. The region has been leftwing since the end of the second world war and is currently led by the PD. At the same time support for the PD and M5S is waning.

“The next key issue will be the elections in Emilia-Romagna and if the PD loses there, I think it would be difficult for them to stick with this coalition,” said Piccoli.

There is a glimmer of hope: Stefano Bonaccini, the incumbent president of the region seeking a new mandate, has been polling slightly ahead of the League candidate, Lucia Borgonzoni.

The Sardines, a protest movement that emerged in November in reaction to Salvini’s threat to “liberate” Emilia-Romagna from the left, might also hold some sway.

Franco Pavoncello, a political science professor and president of John Cabot University in Rome, said if the PD was successful in Emilia-Romagna then the coalition could try to maintain the partnership for a little longer, especially with both parties knowing they would be crushed in general elections if held in the near future.

However, the biggest challenge for Nicola Zingaretti, who leads the PD, and Luigi Di Maio, the leader of M5S, will be keeping their parties intact. A week after the government was sworn in, dozens of PD parliamentarians joined Italia Viva, the group established by the former prime minister Matteo Renzi and more M5S politicians are poised to leave.

“Even if the leaderships understand that the main goal is keeping the far right out, they might simply haemorrhage [more] parliamentarians, that’s when it will become difficult for them,” said Pavoncello.