Discuss the causes for the Mexican-American War, specifically the political situation in Mexico and how the U.S. responded to that situation diplomatically.

The war between the US and Mexico was arguably inevitable, given Mexico's large land holdings following Independence from Spain, the small populations in those regions and the inability of Mexico to get Mexicans to colonize them, and the spread of Americans west. 
Following Independence from Spain in 1822, all of what is today the western and southwestern portions of the US became Mexican territory. The problem was that these areas were sparsely populated (except for Native Americans) and Mexico lacked control over them. Neither did Spain, it should be added, which did what all European powers did during colonization: They drew a line around vast areas of land and then declared it theirs. Mexico City repeatedly tried to send Mexicans to colonize these areas, but many often fled to Mexican cities such as Monterrey to avoid the difficulties of traversing across deserts to an uncertain life in the north. 
Mexico was a turbulent place during this period. Many were still loyal to Spain, some wanted their own Mexican Empire (which briefly existed following independence), while others clamored for a system of government similar to the US. What followed was a long period of political and economic instability.
This was also happening at a time when Americans were pushing west. Mexico realized the problem of having land but no settlers, so they offered Americans land in exchange for becoming citizens and converting to Catholicism. What Mexico did not plan for was that these new Mexican citizens would eventually seek their own independence. This is what happened in Texas in 1836 after years of difficulties with Mexico City over the curtailment of rights and the centralization of authority. Mexico responded by sending General Santa Ana, who was defeated at the Battle of San Jacinto, which then led to Texas becoming an independent country, something which Mexico refused to recognize.
Texas was accepted into the US in 1844 but there were still issues concerning the border with Mexico, particularly as Mexico had warned the US that accepting Texas into the Union would lead to war. In 1846, an American patrol was attacked and an American fort was laid siege to in the disputed territory along the Rio Grande (the present border). The US responded by declaring war, eventually defeating Mexico and briefly occupying Mexico City. The war ended with Mexico ceding more than half of its territory to the US.
The war is still controversial in Mexico, where it is viewed as a war of US aggression. It could be argued that the US was looking for a fight as it eyed the lands of the west. However, there is also the argument as to whether Mexico could rightfully claim the land, given that it gained it from Spain, and Spain's claim was little more than an agreement with other European powers who recognized that claim. In other words, Mexico's claim to the area was simply that they inherited a line drawn on a map by another country on the other side of the globe. Their perpetual problem of physically controlling the land through Mexican colonists led them to enticing American colonists who developed the land in exchange for their allegiance to Mexico. Whether Mexico should have seen the inherent dangers in this exchange, particularly as the US was growing in size and strength, will always be a point of debate.
https://www.history.com/topics/mexican-american-war

https://www.military-history.us/2011/05/th-causes-and-reasons-for-the-mexican-american-war-of-1846-1848/

https://www.thoughtco.com/mexican-american-war-roots-of-conflict-2361034