Bill:
I searched the subscription site Roots Ireland and there were seven 1838 Richard Roche baptismal records in their data base mainly in Co. Cork. Since you don't have parents names, there is no way to confirm any of these records. I would also be skeptical about an Irish emigrant using March 17th as their birthdate. Birthdates including years were not very exact in the 19th century. Age and birthdays were no where as important then as we regard them today. My guess is that your Richard Roche (which was a very common name) was likely born on a different day and year.
I also searched on Roots Ireland for any baptismal records for children of John Cummings and Margaret Kennedy. I only found one record for a Judy Cummins baptized January 26 1844 parents John Cummins and Margaret Kennedy sponsors John Delahunty and Catherine Cummins. The RC parish was Shinrone which is mainly in Co. Offaly but includes a small part of Co. Tipperary.
Records for the parish start in 1842 https://registers.nli.ie/parishes/0760
I would treat this record as a lead. It could be a completely different family.
Let me know if you have any questions.
Roger McDonnell