When you’ve had very little sleep it is hard to deny soft pillows for hard chairs and a key board or other writing implement. If you’re not inspired the words can come out odd and forced together like a group of grouchy three year olds who have missed their naps.
You may even feel like a grouchy three year old who has missed your nap. (I know I do right now.) However, you have an obligation to yourself and to your writing to write. The words don’t have to be pretty just as long as you get them down. If you’re worried you might cause irreparable damage to a current project do a free writing first. (The key phrase here is irreparable damage. This is actually hard to do. Almost anything can be fixed or taken out.)
Or, you can start a new project. If you feel like the grumpy kid that wants to knock over the building blocks, go ahead. Write the absolute worst story you can think of. Have a literary temper tantrum, it will make you feel better.
Then ,when you’re finished, grab you a cup of tea or coffee and settle down with whatever it is you need to work on. Get something done. It doesn’t matter if you can only manage a paragraph or two, that is still progress. It is a lot easier to start the next day if you have managed to get something down the day before.