How many layers are in your file? The white pixels are most likely on a different layer than the one you are working on. A full screen shot of the file in Photoshop with the Layers panel visible will help us be more precise.
Ok, so for a new user this may be the easiest route because it is visual—you can see what is going on—and you start to learn layers in the process:
- Click the Add New Layer button at the bottom of the Layers panel (makes a Layer 2 above Layer 1)
- Edit > Fill > Contents: Black
- Drag Layer 2 under Layer 1
- Click back on Layer 1 to make it active
- Erase the stray white pixels that are now visible against the black background
- Drag Layer 2 to the Trash at the bottom of the Layers panel
P.S. Once the background is cleaned up, you can also work on refining the edges of the text before removing Layer 2. There are places where the blue spills over.
What happens when you either use the Eraser over the spots or use the Lasso Tool to select and then delete the spots? Often, scanners contain dust, which can appear as spots.