There are two ways to go with this:
- Log it as work time and see how it goes
- Ask your boss what you should do and accept the consequences
It is Easier to Ask for Forgiveness than Permission
In many companies, the default response to asking for anything is 'no'. If you ask your manager if you can do something, and they say yes, they are required to take responsibility for the outcome, so they will err on the side of caution and say no.
For that reason, it can be much easier not to ask, use common sense to do what you think is appropriate, and then apologize if it turns out to have been the wrong decision.
If your manager is the type who defaults to no, and this does not seem like the type of issue that would cause problems for your manager if it was revealed, then this is a viable way to go.
Just be sure that if you get caught, you apologize unconditionally, and wholeheartedly. And realize that getting caught may make it less likely to get permission in the future if you ask.
You know boss, I was actually working on work stuff so I don't think that should have been a problem. After all, I was doing something the company asked for.
Sorry for the trouble boss. I thought it wouldn't be an issue but was wrong. I'll be sure to check with you to prevent this sort of thing from happening in the future.
The second is a true apology, doesn't make excuses, and makes the manager more likely to stick up for you in the future because it shows empathy for his situation.
Appeal to Authority
If this may have serious consequences for your boss, and/or your boss is a thoughtful human being who actually considers the request before rejecting it, you can appeal to his more rational side when asking for permission.
Often times when people ask permission, they don't sell the necessity of the thing permission is being asked for. The key to getting permission is:
- Sell the benefit
- Start from 'yes'
Hey boss, I know I'm only supposed to work on that side project for 4 hours a week, but I worked on it 6 hours this week instead. If I don't count the extra 2 hours, I won't meet my 37.5 hours this week. Is it okay if I count those extra 2 hours as work?
Hey boss, I just got in the zone and made a ton of progress on the side project I've been asked to follow! I was so focused, I got 3 weeks of progress done in only 4 hours. I'm going to log that as work time this week, but I'll make up for it next week if required. There's no problem with that, right?
The second sells what you did as the right decision, "Look at how much I got done!". Then it tells him what you will do rather than asking for permission, "I'm going to log that additional time as work time". It also gives him two outs. The first is if someone asks, he can say, "He will make up the time next week". The second is that if he disagrees with any part, he can tell you there is a problem with it.