You're Correct!

A divot refers to the piece of turf that has been dug out of the ground by the golf club when hitting a golf shot. It is important to replace any divots taken to aid the grass growth of the fairway and to ensure that any golfers playing after you do not have to play their shots from a big hole in the grass.

A divot can provide a lot of information to the golfer in both depth and direction. The key point here is direction and to gain information. It is important to stand back behind the ball and review the line of the divot in relation to the line of the target.

Making an assumption that the aim was correct and that the body was pointing correctly at the target, a divot that points:
a) to the left of the target means that the club head was travelling across the ball in an out to in motion (for right handers) at impact. This means that the club will project the ball to the left of the target.
b) to the right of the target means that the club head was travelling across the ball in an in to out motion (for right handers) at impact. This means that the club will project the ball to the right of the target.

Of course the direction of the ball is also determined by the club face of the golf club and so the end point of the golf ball will not necessarily be determined by the club head path that is shown by the divot. What also needs to be taken into consideration is the curvature of the ball in the air. If the ball:
a) curves to the right then the club face was pointing to the right of the club path (divot direction) at impact.
b) curves to the left then the club face was pointing to the left of the club path (divot direction) at impact.

Looking at the combination of divot direction and curvature of the ball in the air it is possible to discover what type of shot was hit and thus work out how to correct that particular shot.

Reviewing the question, a divot that points to the left will initially project the ball to the left of target but what the club face is doing at impact determines what type of shot that occurs. There are two common shots that occur with this divot pattern.

a) a club face that is pointing in line with the divot will produce a straight pull shot to the left of the target.
b) a club face that points to the right of the direction of the divot will make the ball curve to the right and produce a slice shot (for right handers) that begins to the left and curves to finish to the right of target.

Look carefully at the ball flight and divot pattern following each shot to discover faults and evaluate and improve your game.

This may be the case but more often than not it is the swing path that is the problem. Just aiming to the right is a short-term solution and will not fix the problem, in fact you are making yourself swing across the ball more and so only rehearsing the problem.

Not necessarily, the club face dictates where the ball will end up more than the path of the golf club. Look at the two bits of information - the curvature of the ball and the divot before you predetermine where you think the ball will go.

This could be absolutely true but check your aim and swing as the club head path is not always the same as the direction that you aim in.