How did the Industrial Revolution affect the production of this commodity? What, if any, affect did the Great Depression have on the industry?
Other than coffee and tobacco, the main "cash crop" for Cuba (and most other Caribbean islands) was sugar, in the 18th century. This was originally used to make rum for the so-called "triangle trade." (Sugar from the Caribbean became rum in New England or Europe, traded to Africans for slaves, taken back to the Caribbean).
The industrial revolution helped expand this market, because "sugar in tea" was among the first "luxuries" for industrial workers.
The Great Depression hurt this industry, and similar commodities, because the hardest hit people were the above-mentioned industrial workers, who had to cut back on products for which there was inelastic demand. That is products whose prices need to fall more than proportionately to compensate for a drop in demand.
There was a period in Cuba "the sugar revolution" as it is called in Franklin W. Knight's review. Sugar industry includes large set of products, and between 1750-1850 there was a big explosion of demand for sugar. In the beginning of that period aristocracy used up most of the sugar, later in mid 19th century and especially somewhat later even the common people had enough purcashing power to buy some sugar related product. In this period Cuba also lost strict Spanish rule on export-import, and got independent from Spain, introduced and later abolished slavery, by these factors their economy could accelerate.
By 1880's Cuba's sugar production almost entirely ended up in US (according to Wiki, however it is not cited). And we all know US was a serious emerging power in the region by that time.