** Using WeBWorK **

This seems to work -- you do the process the image subroutine outside of PGML. 

The problem, here, is that the string you are trying to pass in $output needs to be exactly a string the you would normally enclose with BEGIN_PGML/END_PGML, but you are getting caught by how Perl handles variable substitution within string literals.

In your example

$a = 1;
$test = "[`$a`]";

the $test string will be [`1`] because the value of $a will be inserted into it before it is assigned to $test. So when you then use [$test]** in the PGML block, this will produce a typeset 1, as you desire. Note, however, that it is Perl doing the substitution of $a, not PGML. If you wanted PGML to insert the value of $a, you would need to use

$a = 1;
$test = '[`[$a]`]';

Note the use of ' rather than " as the string literal quotation mark, which prevents variable substitution. Then the value of $test would be [`[$a]`], and when this is inserted into PGML by [$test]**, PGML will first insert the value of $a and then typeset that value.

The output will be the same, but you should note the difference in the timing of when the value of $a is inserted. This difference is important in your second example:

$output = "

>>[@ image(insertGraph( $gr ), tex_size=>400, height=>400, width=>400,
extra_html_tags=>~~"alt= ~~'$alt1~~' title= ~~'$alt1~~'~~") @]*<<

";

Here, the values of all the variables are inserted into the string before it is assigned to $output. That includes the variable $gr. Since $gr is a perl object, inserting it into a string literal causes it to be stringified, and in this case, you get something like WWPlot=HASH(0x7fc5a6b05f60) as the result. So $output becomes

$output = "

>>[@ image(insertGraph( WWPlot=HASH(0x7fc5a6b05f60) ), tex_size=>400, height=>400, width=>400,
extra_html_tags=>"alt= 'text for accessibility' title= 'text for accessibility'") @]*<<

";

When this string is processed by PGML, the command image(insertGraph( WWPlot=HASH(0x7fc5a6b05f60) ),... is executed, but because $gr as been replaced by a string that is nor part of a valid perl command, the command fails, and no output is produced.

One way around this is to use single quotes as above:

$output = '

>>[@ image(insertGraph( $gr ), tex_size=>400, height=>400, width=>400,
extra_html_tags=>"alt= ~~'$alt1~~' title= ~~'$alt1~~'") @]*<<

';

This will mean that $gr is left verbatim in the string, and so when that is eventually used for a command when processed by PGML, it will properly insert the graphic. On the other hand, it also means that the newlines at the beginning and end of the string are removed by Perl, and since these are critical for PGML, that will prevent the centering from being performed properly (the final << will not be processed without the trailing newlines.

In order to retain newlines, you could use a "here document", as in the following:

$output = >[@ image(insertGraph( $gr ), tex_size=>400, height=>400, width=>400,
extra_html_tags=>"alt= '$alt1' title= '$alt1'") @]*<<

END_OUTPUT

This preserves the line breaks (and also prevents variable substitution), so the centering will work properly. On the other hand, it also means that $alt isn't inserted until the command is executed, and if you are constructing $output using local variables within a function, then that could be a problem (since $alt might no longer be available at that point).

So perhaps you want to use a "here document" that allows substitution, but prevent the substitution for $gr:

$output = >[@ image(insertGraph( ~~$gr ), tex_size=>400, height=>400, width=>400,
extra_html_tags=>"alt= '$alt1' title= '$alt1'") @]*<<
~~n~~n
END_OUTPUT

We need to include the newlines explicitly, since otherwise they are removed again with this type of here document.

Finally, if you are constructing the output string programmatically, you might want to build it up piece by piece. For example

$output = "~~n~~n";
$output .= >[@ image(insertGraph( ~~$gr ), tex_size=>400, height=>400, width=>400,
extra_html_tags=>"alt= '$alt1' title= '$alt1'") @]*<<
END_OUTPUT
$output .= "~~n~~n";

and so on as you add the additional lines of text needed.