Could you please answer my question.
Is sentence: 'Wayne can not spell properly' correct or maybe it should be:
'Wayne can not spell proper'
Thanks for your answer.
The first is right.
The second is common, colloquially, but given the context it would be jocular: 'Some people can't even speak proper'. (The joke is that the speaker is giving evidence that he is one of those people.)
But bare adverbs (without the '-ly') have a respectable history. Read more here: Lessard, Nathalie: The Meaning of the English Adverbial Suffix-ly
Incidentally, there is a collocation where the bare adverb is the only acceptable form; it's fairly informal. 'I messed up the exam good and proper' (='well and truly').