Hi,
Your previous website had some custom CSS added to it in order to add that icon:
monosnap.com/file/rkLrGqlwQfsV28Sc28rJUGdD76BsJR
However, with your new template, there is no such CSS:
monosnap.com/file/veDFibUDR4oDJOkyU8fwOEIh54tCr5
And what's worst is that the classes available in the HTML for the display of the module title (which is done by your template), doesn't allow you to distinguish between the different modules you have.
So even if you add a similar custom CSS there, the icon would appear both for the cart module and the contact MegiQ left module.
So I would recommend to first check with your template provider what can be done to be able to add custom CSS to one specific module with their template. They'll probably have to provide you with some code change in the template files.