So, the verb you are talking about is similar to 'to get' in meaning, but not quite the same. The entire phrase is 'could have had', and you really need to think of it all together, because it is the combination of words that gives the meaning.
This is a modal verb, and these are quite tricky in English.It expresses a possibility that existed in the past, but no longer exists. If he had made different choices in the past, he could have had more time with Katrina. However, the time is over and with the choices he actually made, they spent less time together.
So, let's compare these two sentences:
Now he wishes he could have had longer alone with Katrina.
Now he wishes he could have longer alone with Katrina.
In the first one, his time with her is over. His choices are made. The time they spent alone together is all of the time he will have with her.
In the second one, without the word 'had', he knows he has a set amount of time with Katrina and he regrets that he will not have more, but some of the time they have together may be in the present and some of the time they have together may even be in the future - it just isn't as much as he wants. I think from the comparison you can see that the 'had' expresses that the possibility was completely in the past.
Adding a new example to help explain. Imagine a student, Anna. She is taking an exam right now. She had to work a lot in the days before the exam, so didn't have a lot of time to study. There are a lot of topics on the exam she didn't go over. She is taking the exam right now, so it is too late for her to go back and study them. She wishes she could have had more time to study them.
Now imagine it is Tuesday and Anna will be taking her exam on Friday. She knows the exam will be hard and she wants to study for it. She looks at her schedule and sees that she will be working double shifts on Wednesday and Thursday, so will only have a few hours in the evening each day to study. She wishes she could have more time, but that is all the time she has.
Do you see the difference? In one, all of the time is in the past. She can't get it back. She can't make different choices and get a different outcome.
In the second one, the time period we are discussing is in the future. It is possible (but maybe unlikely) that she could make different choices (for example, quitting her job) that would change the amount of time she has to study.