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Does anyone know of any good surveys on Erdős-Pósa?

I am particularly interested in what are the latest results for the bounding function for directed and even cycles in planar and minor free graphs down to the constant factor.

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I don't know about a survey, but I've found a recent PhD thesis, which seems to be well written:

Heinlein, Matthias (2019): Erdős-Pósa properties. Open Access Repositorium der Universität Ulm. Dissertation. http://dx.doi.org/10.18725/OPARU-11828

The first chapter gives a summary of the problem, known proof techniques and provides references to recent research. I am not an expert on this problem and only skimmed the first chapter briefly, so YMMV.

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