Yes, there was extensive Spanish contact and presence in what is today the United States. In fact, the oldest still-existing European settlement in the US is St. Augustine, Florida, founded by the Spanish in 1565.

It should be noted that while the Spanish did have vast land claims in what is now the United States, the reality of what they controlled was much more limited. The main center of Spanish power, and the region with the most extensive Spanish presence, was along the Rio Grande in northern New Mexico. The Spanish had known of this region since the Coronado expedition of the 1540s, but it was not until 1598 that a permanent Spanish presence was established in the region among the agricultural Pueblo peoples. Spanish attempts at Christianization and exploitation of Pueblo labor would culminate in the Pueblo Revolt of 1680, one of the most successful indigenous revolts ever - the Spanish were expelled from New Mexico and forced south to what is today the city of El Paso/Juaréz and were unable to retake New Mexico until 1692, when the alliances among the various Pueblo groups that had allowed the initial rebellion to succeed had fractured. After this, Spanish presence in the region became less onerous on the Pueblo, with less focus on Christianization and an increasing dependence on other Native groups as an enslaved labor force. Upon American annexation New Mexico had by far the densest non-indigenous population of any of the new territories, and was the only newly-annexed territory where the Spanish-speaking population engaged in armed resistance against the US. The northern half of the state retains a large Spanish-descended population with a distinct local Spanish dialect and history to this day.

Beyond New Mexico, Spaniards explored and mapped vast swaths of North America. The expedition of Juan Ponce de Leon was the first documented European visit to the modern United States, and later conquistadors would make much deeper inroads in the 16th century. The Panfilo de Narvaez expedition to Florida, for instance, ended in tragedy after their ships fleeing Florida wrecked off the coast of Texas. Only four survivors were able to return to Spanish settlements in Mexico after an extensive period of living among the Native population. This expedition was shortly followed by that of Hernando de Soto, a large force which spent several years pillaging and looting their way across the modern American southeast. At the same time, the expeditions of Francisco Vasquez de Coronado ventured across New Mexico and the Great Plains (and were similarly not shy in inflicting violence on the Native population).

Florida would not be permanently settled by the Spanish until 1565, when French attempts at settlement led the Spanish to assert their claims more fully. There was a relative abundance of mission settlements across northern Florida (southern Florida was controlled by the Calusa, who managed to hold off the Spanish), where Native people were forced to settle and take up European-style farming. Disease and abuses led to massive declines in population for the Florida Natives, but the final blow to Spanish rule in Florida was dealt by British-led slave raids at the start of the 18th century. By the end, Florida’s Native population was all but wiped out and Spanish presence had essentially been reduced to the city of St. Augustine. It would not recover before Spain (after briefly losing Florida to Britain) would cede the state to the US in 1820.

Texas was a noticeably different case. After Hernando de Soto, no Spaniards visited for the next century and a half, once again returning only to stop the French from settling. They established missions among the sedentary, agricultural Caddo peoples but saw little success; but eventually, a few permanent Spanish settlements (most notably San Antonio and Nacogdoches) would be established in the state. But the Spanish population was always small and outnumbered by the Natives. This was especially true of the Comanche, horseback warriors who were capable of inflicting devastating raids; countering the Comanche was one of the main reasons Mexico opted to invite Anglo-American settlers into Texas, a decision that would culminate in Texas declaring independence and eventually result in the United States going to war with Mexico.

Spain also briefly controlled French Louisiana, from 1762 to 1803. Again, Spanish presence here was numerically limited, mainly concentrated in the modern state of Louisiana as well as the region around the modern city of St. Louis, Missouri.

I am unfortunately less familiar with Arizona and California, but permanent Spanish presence there arrived later than elsewhere. California had an extensive mission system that was infamous for its brutality, but it was concentrated almost entirely on the coast, with inland groups remaining unaffected.

For further (introductory) sources, I’d recommend:

Knights of Spain, Warriors of the Sun by Charles Hudson, which covers the Hernando de Soto expedition in great detail.

A Land So Strange by Andres Resendez, which similarly covers the Narvaez expedition and its survivors.

The Other Slavery, also by Andres Resendez, covers indigenous slavery under Spain, Mexico, and the United States, with a prominent focus on New Mexico.