How can I add a dbContext after Startup in .Net Core?

You can use an implementation factory overload of IServiceCollection in ConfigureServices method form Startup class:

//First register a custom made db context provider
services.AddTransient<ApplicationDbContextFactory>();
//Then use implementation factory to get the one you need
services.AddTransient(provider => provider.GetService<ApplicationDbContextFactory>().CreateApplicationDbContext());

The implementation of CreateApplicationDbContext depends on your specific needs, but a base implementation should look like the following:

public ApplicationDbContext CreateApplicationDbContext(){
  //TODO Something clever to create correct ApplicationDbContext with ConnectionString you need.
} 

After this implementation, you can inject the correct ApplicationDbContext in your controller, action...

public MyController(ApplicationDbContext dbContext)
{
    _dbContext = dbContext;
}

public IActionResult([FromServices] ApplicationDbContext dbContext){
}

You can always set the connection string from inside the protected OnConfiguring method. You can get access to the IConfiguration instance from there (the DbContext class has a service locator, Instance property), retrieve the connection string, and then call UseMySql extension method with the appropriate connection.

Something like this:

protected virtual void OnConfiguring(DbContextOptionsBuilder builder)
{
    var configuration = (this as IInfrastructure<IServiceProvider>).GetService<IConfiguration>();
    var connectionString = configuration.GetConnectionString("<name>");
    builder.UseMySql(connectionString);
    base.OnConfiguring(builder);
}

For the strongly-typed version of GetService do not forget to reference namespace Microsoft.Extensions.DependencyInjection.