Are you thinking of tsking? I realize that the action isn't the same, but I'm not sure there is a separate word for the specific mouth noise you're describing.
interjection, noun
(a sound) used to express disapproval, genuine or
mock sympathy, etc.: a click, or sucking sound, made by touching the
tongue to the hard palate and rapidly withdrawing it
Perhaps raspberries, as in to blow raspberries:
2 [short for raspberry tart, rhyming slang for fart] : a sound of
contempt made by protruding the tongue between the lips and expelling
air forcibly to produce a vibration; broadly : an expression of
disapproval or contempt
Companies
I will explain to you the position in the United Kingdom which was where the notion of the joint-stock, limited liability company originated, which idea has been adopted in different forms around the world.
The word 'Company' has a legal significance. In the case of a properly registered company the liabilities of the members (shareholders) are limited to the amounts they have agreed to subscribe.
Businesses registered with the Registrar of Companies in this way are called 'Companies'. There are two types. There is the 'Public Limited Company', which has the appendage PLC after its name. These are the largest organisations and the PLC aspect means that their shares are traded on a stock exchange, usually the London exchange. In many cases their total assets run to many billions of pounds. Examples would be BP PLC, or British Aerospace PLC. However some quite small companies are registered as PLC's and their shares are traded. When a small company takes on a stock-market listing it is known as 'flotation'.
The other kind of limited liability comany is the 'private company'. The members' liabilities are limited in the same way as with PLCs, but their shares are not traded. And any transference of share ownership has to be approved by a majority vote of shareholders. These companies have the appendage Ltd., after their names e.g. Norwich City Football Club Ltd, The Delightful Foods Restaurants Company Ltd. etc.
Often large Companies own smaller companies and many are formed into groups of companies. But still the overall group is usually referred to as a 'Company', sometimes as a 'Group'.
Firms
Sometimes the Companies I have described above are referred to, informally, as 'firms', even the very largest ones. The fact that this happens in no way changes the fact that they are legally Registered Companies.
Businesses which are not registered as Compnies come in two forms. Either they are sole proprietorships owned by one person, or they are partnerships. In either case the owners liabilities are not limited. Partners and Proprietors are personally liable, jointly and severally for the debts of the business.
These businesses are nearly always called 'firms'.
Some professional practices e.g. solicitors, accountants etc are not allowed to adopt limited liability under the rules of their professional bodies, e.g. the Law Society or the Institute of Chartered Accountants. These professional practices are nearly always called 'firms'. They also include such as consulting engineers, insurance brokerages, etc. Their names do not have to include the appendage Ltd or PLC.
So, in summary, a 'firm' can mean any type of business, but usually the appellation implies that it is not registered as a company. A 'company' is a business which is registered with the Registrar of Companies and is subject to the Companies Acts.
Corporations
The term Corporation does not have legal significance in the UK, though some Companies are called Corporations, albeit with the appendage PLC or Ltd after its name. Corporation is a term that is far more widely used in the United States. Indeed their equivalent of PLC or Ltd is 'Inc', short for 'incorporated'.
Best Answer
There are several networking terms that can apply here.