A couple things need to happen for Lumumba's government to maintain its position, in my opinion:
- Keep the army from mutiny. Basically everything went to shit when the Leopoldville garrison mutinied. If Lumumba had been able to promise them a pay rise and got rid of reactionary shits like General Janssens the Leopoldville garrison could have stayed loyal and then deployed to quell unrest.
- Get rid of/sideline Kasavubu. Or, more importantly, centralise the government around MNC rather than have a number of different organisations pushing for federalisation. Without Kasavubu and ABAKO, Lumumba could have retained the loyalties of the all-important Leopoldville region and also probably kept Kalonji loyal as well.
With an MNC-lead centralised government with a loyalish army, Lumumba could have quelled unrest in Katanga (maybe) and started building up infastructure and literacy (maybe). There's no way to really stop the power grabs and unrest but at least if he had a way of properly responding the situation might have gone his way and he wouldn't have felt so estranged from the UN and the USA that he needed to go to the USSR in order to get support.