Could you elaborate on what you mean when you say privacy could affect the idea of the client?
The main privacy problem around light clients is: how do you get the client to query the network without revealing exactly what it's interested in? Eg. how do I get transaction information without revealing which are my transactions?
This is the idea behind BIP 157*, which is implemented in Nakamoto. See https://en.bitcoin.it/wiki/BIP37_privacy_problems for issues around BIP37, which is implemented by some other light-clients (eg. bitcoinJ, Bread wallet..)
* BIP 157: https://github.com/bitcoin/bips/blob/master/bip-0157.mediawiki