Words or expressions for Scribbles and Erasure

I've always called this a scratch:

verb (used with object)

• to erase, cancel, strike out, or eliminate (a name, something written, etc.) by or as if by drawing a line through it (often followed by out ): Scratch out the third name on the list.

noun

• a rough mark made by a pen, pencil, etc.; scrawl.

— source: Dictionary.com

This describes both the physical act of scratching out the word, as well as metaphorically erasing it from the text.

On a side note, the phrase chicken scratch might apply if an entire body of text is so poorly written, it approaches illegibility:

An instance of cramped or illegible handwriting

This is often called a cross-out as you suggested. It also could be called a strikethrough or strikeout. While these terms are more often used for printed text that has been deleted, it could be used for handwritten text.

Erasure would likely not be appropriate since it conveys a removing of the offending phrase rather than crossing it out.

When the marks are handwritten, it is often referred to as "crossing outs" or "crossing offs".
If you want to express the action of marking a word or phrase as erroneous then simply say,
to cross out or to cross off.

I'd prefer if you just cross out the wrong answer, rather than go over it with Tipp-Ex/correction fluid.

To delete, erase or rub out a word means that the written word disappears from the page.

In Britain, pencil erasers are normally called rubbers (much to the bemusement of Americans), and I don't think to rub out is a common expression in the US nevertheless, note that the same preposition, out, is used in cross out and rub out.

Scribbles on the other hand, is often used to describe someone's awful handwriting, or when a child is learning to write. You scribble something on a note or on a list especially when you are in a hurry. Usually the only person able to decipher the message is the writer himself.