How to replace, in an XML file, specific strings of characters in some specified tags with other strings of characters embedded in tags

It seems like you want to

1. remove the text an example; from the value of the /c/a node in the XML document, and
2. add a sub-node to the /c/a node called b, with the value a test;.

You can easily do this with xmlstarlet in the shell:

xmlstarlet ed -u '/c/a' -x 'substring-before(text(), "an example;")' file.xml |
xmlstarlet ed -s '/c/a' -t elem -n 'b' -v 'a test;'

The first invocation of xmlstarlet on the example document in the question would result in the following output, where some text is removed from the /c/a node's value:

<?xml version="1.0"?>
<c>This is an example. <a>This is </a></c>

The second invocation takes this modified document and produces the following by introducing the /c/a/b node:

<?xml version="1.0"?>
<c>This is an example. <a>This is <b>a test;</b></a></c>

The xmlstarlet invocations may be combined into a single command. Below, I've used the long options and also use --inplace for in-place editing of the original document (this is for illustration only, you should run without --inplace to determine the transformations works first):

xmlstarlet ed --inplace \
    --update  '/c/a' -x 'substring-before(text(), "an example;")' \
    --subnode '/c/a' -t elem -n 'b' -v 'a test;' file.xml

A generalisation of the above to something that performs the two edits to any a node containing the text an example; (which is what was actually requested in the question):

xmlstarlet ed \
    --var paths '//a[contains(text(), "an example;")]'  \
    --update  '$paths' -x 'substring-before(text(), "an example;")' \
    --subnode '$paths' -t elem -n 'b' -v 'a test;' file.xml 

The only new thing here is that we first store the paths for all the nodes we want to edit in the internal variable $paths. We then refer to these paths in the --update and --subnode modifications.