From what I understand the custom to celebrate a birthday with a cake adorned with candles that makes a wish on and blows out stems from a Greek ritual. As such I would like to know if any halachic authorities have discussed whether or not this should still be considered a heathen custom and therefore prohibited.
Halachically Speaking - page 11 quoting Miyum Hahalacha 4:46 says one should not put candles on the birthday cake.
The practice of putting candles on a birthday cake corresponding to the celebrant’s age does not stem from a Jewish custom and should not be done
Rabbi Eli Mansour quoting Rabbi Avraham Blumenkrantz says it should not be done.
It should also be mentioned that Rabbi Avraham Blumenkrantz, in one of his Pesach digests, documents that the entire concept of "birthday candles" has its origins in ancient pagan customs, and therefore it should be avoided altogether