Amplification of bursts of ultrashort pulses is very challenging when the intraburst repetition frequency reaches the THz range, corresponding to (sub)-ps intervals between consecutive pulses. Periodic interference significantly modifies conditions for chirped pulse amplification (CPA), leading to temporal and spectral distortions during CPA due to optical Kerr nonlinearity. Multi-pulse chirped amplification to mJ energies may lead to a pronounced degradation of burst fidelity and the appearance of periodic temporal satellites after de-chirping the amplified waveform. We study, experimentally and numerically, the limitations of THz burst-mode CPA caused by self- and cross-phase modulation. A number of practical recipes to suppress nonlinear distortions and improve energy scaling by optimizing burst parameters and applying modulation techniques are presented.