You can. Two ways. First, fall back asleep as quickly as possible after being woke. While you fall asleep, keep the dream in your mind. Focus on the details you wish your next dream to follow up on. Change details you didn't like or play out the same scenario in your mind, but a different way or ending. This is scientifically tested, and it's technical name is mnemonic induced lucid dreaming. Though, technically, you wouldn't be lucid dreaming, in this instance. But the method is basically the same as MILD. Do this, and it's likely you'll pick up the dream again when you fall asleep. Ideally, you wouldn't move when you woke up.
Second, and more difficult, is to learn how to lucid dream [awareness of being in a dream, and/or dream control]. Then you can have whatever type of dream you like. Hard to master, though. Takes years and a good deal of dedication and practice. But, the benefits are so huge that it's actually quite small a price to pay, imo. Lucid dreams are just...awe inspiring.
Both scientifically proven. Check BigThink, lucid dreaming, Michio Kaku on youtube for details.
Also, I've found that brown/pink noise helps a lot with outside noises. Not as grating as white noise, too. Sounds like a waterfall.