The date he was wounded per the WW2 hospital admission card is the date he was admitted which is SEPT.
They typically would issue the award to an individual months later. In this case it looks like he was finally award the PH in 4th Dec 1943.
I would advise possible to look for a General Order issued by the Headquarters. You might want to search for it using how typically the Army issued awards through HQ 34th Div, GO 4th Dec 1943.
Please see the Divisions Lineage and view the history. https://history.army.mil/html/forcestruc/cbtchron/cc/034id.htm
Dear Ms. Hendrickson,
Thank you for posting your request on History Hub!
We searched the National Archives Catalog and located the series World War II Operations Reports, 1940-1948 in the Records of the Adjutant General's Office, 1917 - 1985 (Record Group 407) that includes records of the 34th Infantry Division during World War II. We also located the series Unit Histories, 1943-1967 in the Records of U.S. Army Operational, Tactical, and Support Organizations (World War II and Thereafter) (Record Group 338) that includes 23 files related to the 34th Infantry Division during the 1940s. Since the 34th Infantry Division was part of the Fifth Army, the series Award Files, 1943-1945 for the Fifth Army in Record Group 338 might contain some helpful information. The General Order that lists the Purple Heart could be included in these or similar records. The date provided is the date the order was issued. For more information about these non-digitized records, please contact the National Archives at College Park - Textual Reference (RDT2) via email at email@example.com. Please include as much information as possible in your inquiry, such as full name, date(s) of interest, and unit number.
If you have not done so already, we also suggest that you request a copy of his Official Military Personnel File (OMPF). OMPFs and individual medical reports for officers of the U.S. Army who were separated from the service after June 1917 and before 1959 are in the custody of NARA's National Personnel Records Center (NPRC) in St. Louis. In many cases where personnel records were destroyed in the 1973 fire, proof of service can be provided from other records such as morning reports, payrolls, and military orders, and a certificate of military service will be issued. Please complete a GSA Standard Form 180 and mail it to NARA's National Personnel Records Center, (Military Personnel Records), 1 Archives Drive, St. Louis, MO 63138-1002. For more information see Official Military Personnel Files (OMPF), Archival Records Requests. The OMPF may provide a specific General Order and more information about the particular unit the officer you are researching served in.
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 RDT2. Also, NPRC will continue servicing requests ONLY associated with medical treatments, burials, homeless veterans seeking admittance to a homeless shelter, and those involving the VA Home Loan program. If your request is urgent, please see Emergency Requests and Deadlines. Please refrain from submitting non-emergency requests such as replacement medals, administrative corrections, or records research until NPRC returns to pre-COVID staffing levels. Please check archives.gov/veterans for updates to the NPRC operating hours and status. We apologize for this inconvenience and appreciate your understanding and patience.
Finally, the US Army Center of Military History provides some information about the 34th Infantry Division on their website. We suggest that you contact them or the US Army Heritage and Education Center for further assistance.
We hope this is helpful. Best of luck with your research!