First off, i'm not an instructor. So, rather than respond with advice, I am posting this so the instructors here can give me advice about this same problem.
I work on my straight ins at least 1-2 racks each time I step up to the table and I have gone through scenarios like the one op describes. My take on your situation is that the object ball is missing slightly left because the cue ball was shot slightly right.
Therefore, you have an unintended twist in your shot. Others have obviously noted this and recommended fixes but maybe try this: turn your body towards your cue a little bit (if you're a right-handed player).
I feel like I sometimes "square up" too much and then my arm swings towards my body. When it does this, at impact, the cue is pointed slightly to the right (hence missing slightly left). I feel like this was an issue because if I basically face my cue stick (e.g., I turn all the way to the right) my arm tends to swing more to the 'right' in order to avoid my body. While this is an extreme example, I feel like this happens in more subtle ways based on how I am standing relative to the shot.
See if this helps and let us know. Instructors, thoughts on this approach, all other things being equal?