Spain relinquished its claim to Mexico in 1921. I twas more a matter of money and boredom than it was a military loss. Mexico was thought to have huge wealth in gold and silver and jewels. Although some booty was taken back the Spain, it was not the world of riches the Spanish had hoped for. They had better things to do than try to control an uncontrollable rabble that was/is Mexico.

Texas was claimed by the new government of Mexico, although the citizens were never asked. The demand that all Texas resident practice Roman Catholicism and pledge allegiance to Mexico did not sit well with Texans. That shit was just what they or their ancestors had left Europe to get away from.

Both California (in 1846) and Texas (in 1835) declared themselves to be independent republics against the imperial claims of Mexico. Here is a little bit about the Republic of Texas.
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The Republic of Texas

By 1835, Antonio López de Santa Anna had established himself as a dictator in Mexico. Among Anglo-American colonists and Tejanos alike, the call for Texas independence grew louder. On March 2, 1836, a delegation at Washington-on-the-Brazos adopted the Texas Declaration of Independence, and thus was born the Republic of Texas.

Santa Anna had brought his army to Texas to put down the rebellion, and events followed in quick succession. At the time the Declaration was issued, many Texans were fleeing their homes eastward ahead of Santa Anna's army, in what became known as the Runaway Scrape. The Alamo fell to Santa Anna on March 6, and over 300 unarmed Texan prisoners were massacred at Goliad on March 27. Sam Houston's revolutionary army was also retreating eastward as Santa Anna drove for the coast to capture Texas seaports. On April 21, the Texan army took a stand in the bayou country near present-day Houston at a site called San Jacinto. They attacked Santa Anna's army while it was sleeping, and, in a battle lasting only 18 minutes, routed the Mexican army and captured Santa Anna.

Many Texans favored immediate annexation by the United States. However, the proposals went nowhere, because of the risk of continued war with Mexico and Texas' shaky financial status. Even after San Jacinto, Mexico refused to recognize Texas's independence and continued to raid the Texas border. The new government had neither money nor credit, and no governmental structures were in place. Rebuffed by the United States, Texans went about the business of slowly forming a stable government and nation. Despite many difficulties and continued fighting both with Mexico and with Indian tribes, the Texas frontier continued to attract thousands of settlers each year.

In 1841, Santa Anna again became president of Mexico and renewed hostilities with Texas. By this time, sympathy for the Texan cause had grown in the United States, and in 1845, annexation was at last approved. Hostilities with Mexico and the Indians reached a settlement, and Texas was admitted as a state on December 29, 1845. The Republic of Texas, after nine years, eleven months, and seventeen days, was no more.