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I don't have much specific knowledge about this topic, but I can help you try to dig a little deeper into other relevant articles on Wikipedia :)
The two sides of the Spanish Civil War were the Republicans versus the Nationalists. The Republican side constitutes the "left" referred to in the question. The Nationalists sought to restore the "traditional" political order in Spain, which very much included (along with the monarchy and the aristocracy) the power of Catholic Church.
A key even that precipitated the Spanish Civil War was the fall of the monarchy and the formation of the Second Spanish Republic under a new 1931 Constitution. To say that Republicans aimed to "reduce the influence of the Church" in this period is probably an understatement of the hostility. This following is from the article on the history of the Church in Spain:
The Republican government which came to power in Spain in 1931 was strongly anti-clerical, secularising education, prohibiting religious education in the schools, and expelling the Jesuits from the country. In May, 1931, a wave of attacks hit Church properties in Madrid, Andalucia, and the Levant, as dozens of religious buildings, including churches, friaries, convents, and schools, lay in ruins. The government expropriated all Church properties, such as episcopal residences, parish houses, seminaries and monasteries. The Church had to pay rent and taxes in order to continuously use these properties. Religious vestments, chalices, statues, paintings, and similar objects necessary for worship were expropriated as well.
In 1937, two previously distinct factions unified under the Nationalist banner. One of these factions, the Carlist, traced back to the 1830s was always zealously pro-Catholic. Conflicts between liberals and Catholics over land and educational reform were already evident in Spain during that period. The other Nationalist faction in the 1930s was Falangist, which was also pro-Catholic despite some more secularist elements. So while the Nationalist movement and the Catholic Church were far from identical, they were very closely connected.
Finally I will point you toward the article on Catholicism in the Second Republic. There is a lot of detail there about the role of the Church in education, the ties between the Church and rural elites, and so on.
Upvote:1
To answer your question about the Catholic Church in 1930s Spain. The answer is, the Church wanted to avoid change and retain the status quo. To make an overly bold statement, the left were desperate for change based on the model emanating from Russia and the revolution of 1917. This is overly simplistic as there was no single faction of the left, it had varying degrees of extremism from the moderates to the radical POUM who were Marxist and sought change by revolutionary means.The right, on the other hand, were conservative and wanted to retain traditional institutions such as the family and religion. Franco, however wasn't a Fascist (unlike Mussolini) and was opposed to change. One of the identifiers of Fascism is dynamism and this certainly wasn't part of Franco's rationale. The Catholic Church simply wanted to retain the traditional institutions in Spain and preserve its own status.