Check that date, Andy. It is probably 1898, not 1838. The first Mexican Republic (Republica Mexicana) was established in 1822, but they were still using old coin denominations of reales and escudos back then. The Second republic started using the centavos and pesos around 1870.
(Note: If you have a centavo dated before 1869, it might be a pattern or similar experimental coin. Write to us, Andy, if you are sure the date is 1838.)
For 1 centavo coins between 1869 and 1898, the values run about like this:
worn: $2 US dollars approximate catalog value
average circulated: $5
well preserved: $22
fully uncirculated: $75
Coins dated before 1873 are more rare with values about three times these numbers. An 1875 coin with an Ho mint mark under the date is very rare, worth about $400 in worn condition. Likewise coins dated 1872 through 1875 with an Oa mint mark are worth $400 in worn condition.
Use our Important Terminology page to understand how catalog values work. They are inflated values.