There are three main parts to this question.

First, Asturias and Galicia as sources of immigrant laborers. Second, the specific economic realities of Asturias. Third, the draw of Cuba and the impact of its loss on the Spanish economy.

The Northwestern provinces of Galicia and Asturias have been historical exporters of laborers to the rest of Spain. Before the 19th century this was mostly seasonal labor in other parts of peninsular Spain. As the 19th century progressed, however, both saw their surplus population, who were unable to find work in their local economies, go abroad to Cuba and other territories seeking to more permanently improve their standard of living. The gallego grocer is an old trope in Cuban literature. Although less common than those from Galicia, Asturian immigrants also made their presence known in Cuba. These communities also made sure to develop mutual aid societies once they established themselves in Cuba and many did so well that they built enormous 'Centers' in Havana for those of Galician or Asturian descent to visit. The Centro Asturiano became what is now the Museo de Bellas Artes Universal. It should be noted that the Center was built immediately next to Central Park and two blocks away from the Capitol building, which should give some idea of the economic success of these immigrants once they arrived in Cuba.

Between the two, Asturias was better off than Galicia, but still not as booming an economic center as Catalonia or Basque country. Galicia was plagued by the atomization of landholdings (minifundio) after generations of peasant proprietors subdividing their already humble land holdings to the point that they couldn't sustain a family. The general backwardness of Galicia and lack of real hope for improvement spurred many to emigrate elsewhere. Asturias was better off in that its landholdings hadn't gotten nearly as atomized as Galicia (partly due to the fact that most farmers were tenants, not proprietors), its soil was more fertile, tenant policies were more 'enlightened' (important protections for tenant rights, compensation for investments improving the property, etc.), and the increasingly valuable coal mines (which also helped absorb some of the surplus labor that the land couldn't). Nonetheless, Asturias still couldn't produce the jobs and prosperity required to employ and satisfy the needs of the surplus population being produced. As such, many preferred to emigrate either elsewhere in Spain or to the Americas.

In her book Historia de una sociedad en crisis: La Habana a finales del siglo XIX, Maria del Carmen Barcia mentions several Asturians who did quite well for themselves during the late colonial period. One Oviedo banker became a local cacique (political boss) in the expanding sugar region around Cienfuegos and became so rich he bought the aristocratic title of Marques de Cienfuegos in 1893 (five years before independence).

The first two decades of the 20th century were also a major period for Spanish immigration. After Spain lost most of its remaining colonies (the significant ones, at least), the economy went into recession followed by a period of slow growth. While by no means as ruinous as many predicted, the loss of Cuba and other possessions made it difficult for Spanish manufactures used to being competitive thanks to preferential colonial tariffs, bankers who lost major investments in the war, or merchants who lost their special privileges to commercialize Cuban (and other ex-colony) goods. This combined with an already existing economic sluggishness in Galicia and Asturias led to a new flood of Spanish immigrants traveling to Cuba.

Thanks to sugar, Cuba's economy was still expanding and offered many opportunities for capital investment and urban jobs (like shopkeeping). Among those who took advantage of this was a Galician immigrant named Angel Castro, father to Fidel Castro.

The economic crisis of 1920 marked the beginning of the end for Cuban sugar as the country's main cash-cow. Production had finally caught up with and surpassed demand. Prices fell and after the sugar quota of 1934 (the US guaranteed Cuba a percentage of US sugar consumption) those who had invested in sugar began to look elsewhere. Sugar had been a part of the economy you could sink money into and reap enormous rewards. Now, compensation would be received but the enormous growth and fortunes of yesterday seemed to have become a thing of the past).

The 1930s brought a new wave of Spanish immigration from Spain, seeking safety instead of prosperity. The Spanish Civil War (1936-1939) was brutal and saw atrocities committed by both sides. A number of immigrants from Spain's more liberal coastal areas, including Asturias, fled to Europe, the United States, and Latin America. As Franco's side won and his regime ruled until the 1970s, many ended up building lives abroad and not returning to Spain after his death and the rapprochement attempted in the post-dictatorship years.

Bibliography:

Raymond Carr: Spain, 1808-1875 (second edition), Oxford University Press, 1982.

Maria del Carmen Barcia: Historia de una sociedad en crisis: La Habana a finales del siglo XIX, Editorial Ciencias Sociales, 2000.