Explain in your own words how the South expected cotton to influence recognition of their side by Great Britain and France during the Civil War.

The Confederate States were eager for recognition by a foreign power during the Civil War and their leaders hoped that cotton would help secure that. Foreign recognition could potentially ease the blockade that the United States Navy used to bottle up the breakaway slave states. The blockade reduced their access to weapons, iron and other products needed to carry out the war.
Using slave labor, the South was a major producer of cotton, which was converted into textiles in industrialized England and France, powering their economies. Confederate leaders hoped those nations' reliance on cotton would prompt them to intercede on the South's behalf. However pre-war exports of cotton from the southern states led to surpluses overseas, reducing the demand after the outbreak of fighting in 1861.

In the 1800s, one of the main materials traded to Britain and France from the United States was cotton. This cotton would then be utilized in factories in Britain and France to produce textiles which could then be sold in their own countries, in their colonies, and elsewhere around the world. The cotton that was traded to Britain and France came from the southern United States, where it was grown and harvested using slave labor. One of the reasons cotton production was so profitable for the South was the fact that production costs were relatively low because of slavery. This would also keep prices low for the British and French who hoped to purchase cotton for production of textiles.
In the 1860s, Britain and France had already banned slavery. Morally, these countries were opposed to slavery. The South hoped, however, that the economic benefit of trading with the Confederacy would cause Britain and France to question just how valuable the morals were.
https://www.pbs.org/wnet/african-americans-many-rivers-to-cross/history/why-was-cotton-king/