Question: I was trying use span and style tags in pages, rather than in CSS and it seems that Drupal is configured to strip them out whether the input format is *Filtered HTML*, i.e. they are applied to *Full HTML*.

Is that something which can be fixed only by editing the core files, or are the some modules or settings in Drupal's admin settings that can override that?

Answer: 
Are you using a WYSIWYG module like "CKEditor - WYSIWYG HTML"?

It appears to be this module or the CKEditor itself that is wrongly stripping out correct HTML such as inline styles, not Drupal.

There is a bug report open here <https://www.drupal.org/node/778200>

My current workaround is to disable the editor when adding these types of HTML so that I edit and save while in source mode.

You could also try using a different WYSIWIG editor besides CKEditor. This module allows several different editors to be used: <https://www.drupal.org/project/wysiwyg>
To configure Text Formats you can go to /admin/config/content/formats 

> 
> A text format contains filters that change the user input, for example stripping out malicious HTML or making URLs clickable. Filters are executed from top to bottom and the order is important, since one filter may prevent another filter from doing its job. For example, when URLs are converted into links before disallowed HTML tags are removed, all links may be removed. When this happens, the order of filters may need to be re-arranged.
> 
> 
> 

See [here](https://www.drupal.org/node/213156) for more information on text filters and input formats.