Kubernete have no access from eks pod to rds mysql
I did had the same scenario (RDS in the same VPC as the EKS cluster). What I did is as following:
I've created a Cloudformation template with which I created my custom VPC, 8 subnetes(3 public, 3 private for EKS cluster and 2 private networks for RDS database), internet gateway, NAT Gateway, route tables and routes.
Using eksctl with cluster configuration yaml I created the cluster and the node group. The node group joined my cluster.
Using aws cli, I've created the db-subnet-group (containing the 2 private DB subnet) and I also started and RDS instance. Then I've set up some security group to allow traffic to DB just from the 3 private subnets)
As reference to create my custom cloudformation template I used the template created by eksctl when running the create command with the flag --node-private-networking.