GridsearchCV in Python
I assume it is machine learning model (e.g. neural net)

When running lets say CVfold =10 in gridsearchCV, the model is getting
trained on 9 partitions and tested in the remaining 1 partition.
the avg auc results coming from the 9 partitions or the avg

Usually he model is trained on 1 partition (train set) composed from 9 arbitrary partitions. Therefore there is not such a thing as avg AUC from 9 partitions, there is only one train AUC. This words are not true if you are sure that you train on 1 partition, calculate metric, train on 2nd, caluclate metric and so on until you have got metrics results from 9 partitions and averaged them.
The key question:

Question is what is more relevant here?

Depends what is the question you are answering to.
Results from test partitions should tell you about what performance can you more or less except when you release the model to the world (make predictions on unseen data). However it is easy to introduce some sort of data leakage when you use CV, data leakage makes the results less trustful.
Comparison between training and test should tell you if you are overfitting your model or if you should make model more fitted. I do not rule of thumb of how much the difference is fine (suggest further reading about overfitting) but I have never seen anyone accepting 20% difference.
I have a conceptual question and hoping someone can clarify. When running let's say CV=10 in GridSearchCV, the model is getting trained on 9 partitions and tested in the remaining 1 partition.
The question is what's more relevant here? The avg AUC results coming from the 9 partitions or the avg AUC of the testing partitions. What if the AUC's on these 2 (9 vs 1 partition) are far apart, let's say more than 20% apart. What does that say about the efficacy of the trained model? Any rule of thumb on how far the AUC's could be? What is generally reported as the measure of model performance, the 9 partition AUC (train) or the testing partitions?
Thank you!