The earliest English Drama that we note as wholly English would be the Mystery, Miracle and Morality plays of the Middle Ages. Though they were meant as teaching tools of the Church, meant to convey stories and lessons of Christianity to a largely illiterate populace, they, of necessity, bore the marks of secular influence, most notably in the use of theatrical device.
These plays, in many ways, need to be considered as secular events because of the context in which they would be performed -- NOT in a church. Wagons traveled to an open area -- an inn courtyard or open street -- opened their doors and performed. It was the necessities of a theatrical rather than liturgical environment that sparked the secular influence.
Actors want to keep an audience engaged, and one of the easiest ways to do that is through humor. So a great deal of the secular investment into these plays was through comic actors inventing action and text which diverged from the Biblical story or moral being presented. Comic actors or clowns, were know all the way up to Shakespeare's day for their ability to improvise humorous action and remarks, sometimes to the detriment of the plot of the play.
The Vice character of Morality plays was another ultimately secular influence. Though meant to show the audience the downfall of behaving badly, Vice was often a funny, wily character, whose charm proved very seductive to the audience he was meant to repel. Shakespearean villains like Iago and Richard III are direct descendants of the Vice character.
Though there are other influences, I would suggest that humor and the power it affords an actor in winning and maintaining the audience's attention is a major secular addition to Early English Drama. More secular influences can be found at the links given below.