Incorporating known ciphertext into new message Which ciphers $c_k$ enable one to define a joining function $f_{c,k}$ such that:

$$
f_{c,k}(c_k(p_1), p_2) = c_k(p_1 + p_2)
$$

That is, the result of joining some known ciphertext $c_k(p_1)$ with a plaintext $p_2$ produces the same ciphertext as the concatenation of the respective plaintexts?  Whilst there is some flexibility to modify $p_2$ (e.g. with padding) it cannot be chosen entirely arbitrarily.

What terminology applies to such cipher/function/messages?
Not sure if this fits your requirement, but, here goes.

Using the homomorphic properties of ElGamal, you could turn $E_{pk}(b_0),E_{pk}(b_1),E_{pk}(b_2),E_{pk}(b_3),E_{pk}(b_4),\cdots,E_{pk}(b_n)$ (where $b_i\in\{0,1\}$, i.e., individual bits), into $E_{pk}(b_0||b_1||\cdots||b_n)$ (where $||$ is concatenation). This is done by using homomorphic scalar multiplication and addition (multiply by 2 then add the next bit, repeat as necessary).

This could be generalized to larger messages (not just individual bits) if Trent is told the lengths of the messages.

Any cipher that is homomorphic with respect to addition could be used.