LOL, I don't when I told zeusky bhai about Kohli's abdomen guard

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If you take Micheal Clarke as an example then in his case what happened was that he left Slazenger after first test match and during the middle of the series. So second and third test he played without sponsors while in fourth he got a sponsor (Spartan). From what I have heard Micheal Clarke is extremely picky about his gear, so for any person new pads and gloves take time to break in and mould himself. So rather than using new gear what he did was continued to use his old sponsor gear as he was most comfortable using them compared to new sponsor softs. Second reason could be he simply didn't like Spartan softs so he didn't like it, however, Spartan will be launching a range called 'MC 329' and it will be endorsed by Micheal Clarke. So MC 329 range softs could be how Micheal Clarke want his softs to be.
The meaning of sponsorship is advertising a company. Everyone in this world is after money, so having said that international players will go the company that will pay them the most. For example, there are rumours which say that Micheal Clarke is being paid nearly a million each year by Spartan for three years deal, that is much more than what Slazenger paid him.
Now answering your question where you said people use different brand softs. For example, Asad Shafiq is sponsored by RNS but he uses SS softs. This could be purely based on preference, he prefers SS softs over RNS softs for various reason such as comfort being the most important one.
Why they cover the logos? A sponsored player for a specific brand is being paid by that company to promote and advertise them. If a player is wearing pads or gloves which are not of his sponsor company then he has to cover the logos because if he doesn't then it means he is advertising that company while being paid from a different company. For example, Hashim Amla is sponsored by BAS so if he wears SS pads then reason could be that they might be more comfortable for him compared to BAS pads. Now he has to cover logo of SS pads because SS is paying him to wear there (BAS) pads and if he goes out and play in SS pads (without even covering the logo) not only he is giving SS free advertisement but he is breaching rules of his BAS sponsorship and he could be fined or his sponsorship could be dropped (happens in some cases).
Hope this all make sense