Present, unrepentant sin is never covered by Jesus' atonement- in the sense that it is forgiven. But obviously there is an atonement available for sin that a person who will deal with their sin as they ought to by confessing and forsaking it with a real living faith in the Jesus of the Bible can have applied to them (see 1 John 1:5 to 1 John 2:6). Obviously this would be true of the sin of unbelief also. There is no one who has never thought nor acted in unbelief who doesn't need pardon for that "for whatsoever is not of faith is sin." (Romans 14:23b)
Now when it comes to the blasphemy of the Holy Spirit, I suppose it's logical to say that Jesus' work on the cross would not, as well as could not, cover that. But it is also logical to conclude that this is no way validates James White's theology since the Bible is clear that God is no respecter of persons. By that I mean that ANYONE who truly blasphemes the Holy Spirit (and most who think they have done so in fact have not) would be beyond the hope of pardon; and no one who has truly done so had to do so (they could have refrained and also repented instead). There is no validity to limited atonement then just because someone one may be beyond pardon, since such people didn't need to commit the unpardonable sin; and before they had committed it they could have repented and found pardon through Christ's blood. So overall Christ's atonement not being able to cover the unpardonable sin only proves there is a sin the atonement doesn't cover; it by no means proves (as Mr. White believes) that there are people which Christ's atonement could never have possibly covered and been applied to. White is using a strawman argument which is even worse because his own position is so obviously refuted many, many times in the Scriptures.
Look at the "all"s in this passage: 1 Timothy 2:1-6 "I exhort therefore, that, first of all, supplications, prayers, intercessions, and giving of thanks, be made for all men; 2 For kings, and for all that are in authority; that we may lead a quiet and peaceable life in all godliness and honesty. 3 For this is good and acceptable in the sight of God our Saviour; 4 Who will have all men to be saved, and to come unto the knowledge of the truth. 5 For there is one God, and one mediator between God and men, the man Christ Jesus; 6 Who gave himself a ransom for all, to be testified in due time."
And then the nail in the coffin of Mr. White's argument and theology:
Romans 11:30-32 "30 For as ye in times past have not believed God, yet have now obtained mercy through their unbelief: 31 Even so have these also now not believed, that through your mercy they also may obtain mercy. 32 For God hath concluded them all in unbelief, that he might have mercy upon all."