How to secure Elliptic Curve ElGamal encryption against known plaintext attacks?
Is there any way to compute a different $R$ for every block of data, without using much resources and that too quickly ?


There's no common way. Standard practice is hybrid encryption per ECIES.

In a nutshell, ECIES is the same as EC-ElGamal with regard to $E$, $G$, $\text{privatekey}$, $Q$, $k$, $C$, $R$, but the shared secret $R$ is used (after a key derivation step) as the key to a symmetric authenticated cipher that conveys the message $m$. This removes the burden of mapping $m$ to a point on the curve, which severely limits the size of $m$, and typically is iterative thus has the potential to leak information about $m$ by side channels.


  Does reusing the same $R$ for another message break security (in EC-ElGamal)?


Yes. Assume $m_0$ and $m_1$ mapped to $P_0$ and $P_1$ are encrypted with the same $R$, into $C_0$ and $C_1$. It holds $C_0=R+P_0$ and $C_1=R+P_1$, therefore $P_1=P_0-C_0+C_1$. Thus $m_1$ can be found from $m_0$ and ciphertext. With $R$ fixed, the cipher is insecure under known-plaintext attack.

Additions per comment:


Even if the attacker has no knowledge of a plaintext $m_0$, s/he does learn something about the plaintext. In particular she can tell if $m_0=m_1$, since that's equivalent to $C_0=C_1$. This qualifies as a break, since the objective of encryption is to prevent adversaries from learning anything about plaintext (except its length). As an example, this allows to distinguish between a routine Have a quiet nightshift Joe ($m_0=m_1$) from an exceptional Launch missile to target A ($m_0\ne m_1$).
Doubling $R$ at each message would be extremely insecure. It would hold $P_j+P_j-P_{j+1}=C_j+C_j-C_{j+1}$, and that allows to decipher any message sent encrypted twice, as $P_j=P_{j+1}=C_j+C_j-C_{j+1}$.