Will running games like hayday or clash of clans in the background for a very long time consume battery?
In iOS if an app does not do background processing it does not take up battery. Games should not be doing background processing.

When you get low on memory iOS will send a signal to all apps so that they can reduce their memory usage and some will free memory. If there is still not enough memory for the current app iOS will terminate other apps.

Thus iOS will terminate apps for you - you do not need to terminate them manually and there is no benefit in doing so.

So leave the apps alone..

Of course if an app is badly written and does background things then terminating might be needed but this is unusual and the developer has to do this deliberately so I thin k games should definitely be OK - there are some apps that look at your location which do misbehave but hames don't need this.