CAP Theorem and MySQL
What you are proposing is not really CAP Theorem unless the cluster is totally a share-nothing architecture. Please note what CAP Theorem is from my old post from Jan 01, 2013 : Consistency in ACID and CAP theorem, are they the same?
CAP

Consistency (All Nodes Have Same Data via Eventual Consistency)
Availability
Partition-Tolerance : system continues to operate despite arbitrary message loss or failure of part of the system

Each node would have a complete copy of the full dataset. The given DB Cluster would need to be support eventual consistency i.e. have ACID Compliance support cluster wide.
YOUR ACTUAL QUESTION

But is it true in fact of using MySQL? Or maybe we need to use any kind of middleman? How does CAP theorem apply to MySQL?

The MySQL setups that operate on CAP Theorem are as follows:

InnoDB Cluster
MariaDB Galera Cluster
Percona XtraDB Cluster

Your best middleman ??? If the data you write is critical and needs to be available, read the data from the Master you wrote it. In any of the aforementioned Cluster Setups, any node can be a Master. If data is written on one Master, they should all have it and be ready to be read back provided all nodes in a Cluster run COMMIT on that same given transaction.
If you have slaves attached to these Masters, each slave must wait on its Master COMMIT for data to be be replicated and become visible on a slave.