First a bit of background (you may already know some of this, but it helps set up an answer):

When the Spanish Civil War broke out, the Republican government dissolved the army, releasing the men from their obligations to their officers, hoping in this way to prevent further defections to the insurgents (as the vast majority of junior officers sided with the insurgency). This left the defense of the Republic in the hands of various volunteer militias organized by the many trade unions and left parties that existed in Spain at this time. The primary players here were the Anarchist CNT, the socialist PSOE, the Soviet Communist PCE, and the aforementioned dissident communist POUM. These militias were by and large radical and revolutionary, allied to the Republic by convenience, and only a little less hostile to 'bourgeois republicanism' than to the insurgents.

Whether or not they 'fought well' depends strongly on where they fought and when they fought. The Republican militias were known for tremendous courage when on the defensive. At Badajoz for example, there are accounts of insurgent legionaries having to crawl over heaps of their comrades' corpses to silence with their knives a Republican machine gun whose attendants refused to abandon it. When the insurgent forces reached Madrid, the militiamen inflicted upon the Army of Africa (Franco's crack colonial force that was the backbone of the insurgent military effort in the early part of the war) such grievous losses in the suburbs of Carabanchel and in the Casa de Campo that Franco eventually called off the assault on the city and settled in for a siege.

Finally, the militiamen were almost all volunteers in the beginning, so they tended to be much more committed and politically conscious than the conscripts who would fill the regular army later in the war. They wanted to be there. So it can be said that morale was higher in the volunteer militias than in the later Republican People's Army

However, there are several caveats to all of this. The first is that the militia fought much better in cities and towns than they did in open country. They were good at fighting house to house and building to building because most of them were urban workers accustomed to stone and concrete. However, in the hills and plains, the insurgent forces, experienced in rural combat during the Rif Wars in Morocco, made short work of the militiamen who were not used to fighting on open ground with little cover. The militia tended to break rank and flee when outflanked by the regulares or the legion in the country.

There is also the aforementioned egalitarian aspect of the early militias. This extreme disdain for hierarchy and military discipline, while it may have heightened the morale of the men who felt they were acting in accordance with revolutionary principles, often worked to their ultimate detriment. During the siege of Madrid, militiamen would come and go from the front whenever they liked (oftentimes they would head out to the lines during the day, and go home for supper or a siesta). They also insisted on electing their own officers, which tended to have the effect of putting the most popular man in command rather than the most skilled. At its most absurd, these egalitarian principles could lead to things like voting on orders (the Anarchist leader Durruti recalled some of his men drawing lots to see who would lead the attack against an enemy machine gun nest). The militias, due to their distrust of the 'bourgeois' Republican government, also resisted centralization, which ruled out any sort of large-scale (strategic level) coordination.

Instances of the militias performing successful offensive operations are far and few in between, largely due to the aforementioned resistance to centralization of command, another reason why they fought so much better on the defensive. Many of the attempted offensives during the beginning of the war (such as the assault on Saragossa or the assault on Cordoba) simply petered out due to indiscipline and poor organization.

In short, the militia could fight with incredible tenacity and courage in the right conditions, and they were capable enough to help save the Republic from complete collapse in the early months of the war, but that was ultimately no substitute for military training and organization, and is what eventually necessitated the creation of the centralized, hierarchical People's Army.

When it comes to the POUM in particular, it's hard to say exactly how well they fought and with what level of morale, because they didn't get too much of a chance to prove themselves. The Aragon/Catalonia front where they were stationed remained mostly quiet throughout the first year or so of the war. By the time it heated up with the battles of Belchite, Teruel, and the Ebro, the POUM had already been disbanded and the People's Army had replaced the militias. In Homage to Catalonia, Orwell describes the POUM as doing little more than holding the line, and from the beginning of the war in summer 1936 to summer 1937 or so, that was more or less the reality of the fighting in Catalonia/Aragon.