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: 
Go to Configuration -> Content Authoring -> Text formats. Here you can configure what tags are allowed.

All unallowed tags are stripped out, is always stripped out.

Check out <https://www.drupal.org/project/wysiwyg_filter> for more advanced filtering.
To configure Text Formats you can go to /admin/config/content/formats 

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> 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.