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Here is some information (click on images to enlarge). Here is where he is burried. https://www.findagrave.com/memorial/8921481/william-ashbury-haston
Dear Mr. Haston,
Thank you for posting your request on History Hub!
We searched the National Archives Catalog and located the Indexes to the Carded Records of Soldiers Who Served in Volunteer Organizations During the Mexican War in the Records of the Adjutant General's Office, 1762 - 1984, (Record Group 94) that includes an index card for Haston, William A - State: Missouri - Regiment: 1 Missouri Mounted Infantry, Company G - Enlistment Rank: Pvt - Discharge Rank: Pvt. The card has been digitized and is available via the Catalog. We also located the Carded Records Showing Military Service of Soldiers Who Fought in Volunteer Organizations During the Mexican War in Record Group 94 but were unable to locate a digitized file for William Haston. Most of the service records in this series have been made available via microfilm and through digitization of the microfilm. Unfortunately, the records for the Missouri Volunteers have not been microfilmed or digitized. For information about the non-digitized records, please contact the National Archives at Washington, DC - Textual Reference (RDT1) at email@example.com.
In addition, we located the Case Files of Pension Applications Based on Death or Disability Incurred in Service between 1783 and 1861 ("Old Wars") and the Registers of Pension Certificates Issued to Army and Navy Veterans Between 1816 and 1914 in the Records of the Department of Veterans Affairs (Record Group 15) but were unable to locate a file for William Haston. These records have not been completely digitized. To request a search and order copies of military pension application files based on Federal (not State) service during the Mexican War, please complete a NATF Form 85 and mail it to the address on the form.
Due to the COVID-19 pandemic and pursuant to guidance received from the Office of Management and Budget (OMB), NARA has adjusted its normal operations to balance the need of completing its mission-critical work while also adhering to the recommended social distancing for the safety of NARA staff. As a result of this re-prioritization of activities, you may experience a delay in receiving an initial acknowledgement as well as a substantive response to your reference request from RDT1. Please note that the National Archives and Records Administration (NARA) has suspended reproduction and digitization services until further notice due to COVID-19. Orders will not be serviced until operations can resume safely. Once operations resume, document reproduction requests will be filled in the order in which they were received. We apologize for this inconvenience and appreciate your understanding and patience.
We hope this is helpful. Best of luck with your research!