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.