J’ai la course à faire does not sound like proper french.

J’ai une course à faire can indeed be used to say go shopping but has a more broader meaning. For example, it’s ok to use it when you go to the postoffice, to the hairdresser, if you have to go pickup something,…

It’s also quite a generic expression that can be used as an excuse to leave if you do not want to say specally where you are going.

Please provide more context if you want a more precise answer.

Based on @Ndech’s response, it seems the most accurate translation of j’ai une course à faire would be I have an errand to run. It is possible then, if the character on the TV program in your question referred to the errand in an earlier sentence, that they could use la course to refer to the errand previously mentioned.