Log Shipping with Failover Cluster Server Cores Failover Over Cluster 2 Node - 6 Cores- Enterprise Edition 2014 
Log Shipping with Cluster with 4 Core License- Enterprise Edition 2014 


My question is if I need to log ship do I need to match the server core of the Primary server that is the failover cluster (6 core) in my case or I can go with 4 core on the Log Shipped server to save licensing costs?
No, you don't need to match hardware. There's no technical requirement for that.

Lots of people choose to use slightly less robust hardware at DR


Saves on licensing costs
Offloaded reads may not need full production power
Assume that DR failovers aren't permanent


That said, whatever lower tier of hardware you choose has to be able to support your current workload. 

In general, you want to make sure you're not hitting any really bad resource waits like THREADPOOL -- if you're hitting it on 6 cores, you'll definitely hit it on 4 cores.

If you have less RAM at DR, be aware of RESOURCE_SEMAPHORE waits. 

The only way to tell for sure is to coordinate a failover and measure your server's response to your workload.

Hope this helps!