Adding tweak to a block cipher I know there are XEX, XTS and other ways to add tweak to block cipher without modifying cipher itself. However they are quite slow and/or complex.

If we assume we have a secure block cipher round function (like AES) with pseudo-independent round keys (generated with stream cipher). Is it safe to add tweak with same size as block directly (and same value every round) into round function the same way as round keys?

$F()$ = round function
$b$ = block
$k_r$ = round key (same size as block)
$t$ = tweak (same size as block)  

$b = F(b)$
$b = b \oplus k_r \oplus t$

How strong tweak schedule should be?
This construction isn't generically secure, you need to analyze it for each blockcipher you want to use it with to see if it's secure.

For example, consider a block cipher that simply xors the key into the state between rounds. In that case your construction is equivalent to xoring the tweak into the key. This has several consequences:


Since we generally assume related and attacker controlled tweaks, this exposes the blockcipher to related key attacks. There are blockciphers which are secure with random keys, but not related keys (e.g. AES)
It enables multi-target attacks, with each tweak acting as a separate target. For example if the attacker sees a 4TiB of known plaintext, that corresponds to $2^{42-4}$ 128 bit blocks, reducing the security level of a 128 bit cipher to 90 bits.


Even if there is no direct interference between the cipher's key schedule and the tweak, the tweak gives an attacker more influence over what happens inside the cipher (assuming random keys but attacker controlled plaintext/ciphertext/tweak). To compensate for that, you need to increase the number of rounds, decreasing performance.

=> This construction might be worth a consideration when designing a new tweakable blockcipher to replace AES, but it's not something you should simply add to an existing cipher, like you can with XTS and similar constructions.