Wear on?

[bartholomew]

Because of Texas dialect coupled with mangled words, Vernon's narration initially wears on the reader.

[rampart]

The reader of this narration will not like the Texas dialect and mangled words at first. But I deduce from your sentence that as the reader continues, the dialect and mangled words bother the reader less and less. The phrase "wears on the reader" is kind of like "annoys the reader."

Think of you on a bus and across is a lady whose kid screams and cries and the lady does nothing. This would "wear on you."

Hope that helps ~

[Artful]

As rampart says:

to wear on (your) nerves = to get on (your) nerves,  to annoy you more and more. The Chambers Concise defines it as to become oppressively irritating.

[jno]

'Wear' means to annoy or just to tire (weary means tired).