The problem is you already have a version of some MySQL packages installed that are from a different distribution/vendor than CentOS.
Since different people created the packages using different names and standards, yum does not know how to fix this situation.
You have libmysqlclient16-5.1.69-1.w6.x86_64 installed already providing the client libraries, but the CentOS package is mysql-libs. This package appears to be provided by "webtatic":
https://webtatic.com/packages/mysql55/
http://rpm.pbone.net/index.php3/stat/4/idpl/21966562/dir//com/libmysqlclient16-5.1.69-1.w6.x86_64.rpm.html
Likely you need to consider removing the libmysqlclient package and installing mysql-libs in it's place. That will probably remove some items because of dependencies, you might be able to get away with manually removing it with the rpm --nodeps -e, but it may also be the packages cannot have their dependency satisfied by mysql-libs so slightly risk move if you don't fully check everything first. This often works, though.
As an alternative looks like you can install the mysql server package from webtatic as well, based on their instructions here, that might be a less complex alternative if you are happy to continue relying on their repository:
https://webtatic.com/packages/mysql55/