HD wallets with an exposed child private key HD wallets as described here are built based on a master seed and a master public key as follows:

PK.b = PK.a + sG
SK.b = SK.a + s



I just got a little confused about this. Is s the mnemonic seed that
user knows? which generates the private keys. 
Secondly a service
provider only needs to know sG to calculate the user's next address.
Am I right? 
And lastly more importantly I am just wondering whether it is possible to
calculate the master private key (seed) if an attacker or service provider gets one child's private key address?
I don't know where you found this formula, here is the code to derivate a childkey, the BIP is here, it does not look like your formula, but I'll try to respond to some of your questions.

I just got a little confused about this. Is s the mnemonic seed that user knows? which generates the private keys.


A mnemonic seed is just a sentence that permit you to get the master key with (optionally) a password. A master key is the real important piece of information, a mnemonic is only a way to generate one that can be write it down in words.

Secondly a service provider only needs to know sG to calculate the user's next address. Am I right?


Your formula does not look like what is in the BIP, so I can't really comment on that.

And lastly more importantly I am just wondering whether it is possible to calculate the master private key (seed) if an attacker or service provider gets one child's private key address?


Yes, if all the children in the path to the leaked private are non-hardened keys AND the root HD PubKey leaked. I explain that P 35 of my book.

A non-hardened key can “climb” the hierarchy.
Non-hardened key should only be used for categorizing accounts that belongs to a single control.