I'm following <PERSON> direction of thinking.
1. Mars and Earth orbit the Sun. A single blow can't put Mars into path where it would gradually decrease its distance to Earth for 300 years.
2. One could attempt to put Mars onto an eccentric orbit intersecting with Earth's orbit. That would require a knock of velocity of at last 2500 km/h (calculations below), thus quite some more than 21 km/h. Mars would then intersect the Earth's orbit in less than a year. Depending on the initial positions of the planets on their orbits the collision could occur during the first intersection or one of the further ones. If however they would not collide for 300 years, the Earth's orbit could be still perturbed by the gravitational force of Mars, which by itself could have catastrophic results. Predicting the details would require a complex simulation.
3. If you want to postpone the collision for 300 years, I would propose to knock Mars away from the Sun, so it would cross Neptune's orbit and come back after 300 years to hit the Earth. One still would need to take into account possible interactions of Mars with the outer planets and perform very precise calculations, but that might be easier than keeping Mars just next to the Earth without a collision for 300 years. The knock velocity to the planet would need to exceed 11500 km/h so even more. An advantage of this method is that depending on the planets position the recoil may put the protagonist into virtually any spot in the solar system you want, including the Earth or its Moon.
4. As the others pointed out, if the protagonist transfers momentum to the planet, it receives the same momentum in the opposite direction. For 2500 km/h knock on Mars a 100kg protagonist would receive energy of 16·10¹⁵ GeV per each nucleus of his body. 921
This is a lot. That's way more than any energy ever produced in laboratory or observed in nature. That's a hypothesized energy of Grand Unification (https://en.wikipedia.org/wiki/Grand_unification_energy). The act of knocking would be extremely violent. Complete disintegration of any molecular and nuclear structures, production of any known and perhaps unknown particles, maybe even black holes.
5. Some suggested that the knock would destroy the planet. Maybe. The energy of the knock described in point 2 is 39·10³⁰ J, while Mars binding energy is perhaps 5·10³⁰ J. Some matter would follow in the desired direction. It's difficult for me to speculate how exactly it would turn out.
6. Once the protagonist lands on another target planet or moon, it transfers its momentum to it, causing similar level of destruction and orbital perturbation as that done to Mars. As the others pointed out, it would violently interact with any matter on its way, and even with cosmic background (see: https://en.wikipedia.org/wiki/Greisen%E2%80%93Zatsepin%E2%80%93Kuzmin_limit).
Maths. I'm sorry if there are any mistakes, but note we're in the regime where adding or deleting a zero wouldn't change the conclusions.
1. Knocking Mars towards the Earth The total (potential and kinetic) energy of a planet with mass m orbiting around a star with mass M is $$E = -G\frac{Mm}{2a},$$ where a is the semi-major axis.
Assuming Mars following a circular orbit, $a=R_\mathrm{Mars}$ (distance between Mars and the Sun), $$E_\mathrm{Mars} = -187·10^{30} \mathrm{J}.$$
For elliptic orbit between the present Mars and Earth's orbits $a = (R_\mathrm{Mars} + R_\mathrm{Earth})/2$, $$E_\mathrm{Mars-Earth} = -226·10^{30} \mathrm{J}. $$
The knock corresponds to change of the energy by $$\Delta E = E_\mathrm{Mars-Earth} - E_\mathrm{Mars} = -39·10^{30} \mathrm{J}.$$ As the potential energy remains constant, $\Delta E$ corresponds to change of the kinetic energy only.
The orbital velocity of Mars is $v_\mathrm{Mars} = 86430$ km/h, it's initial kinetic energy is $E_\mathrm{k} = mv_\mathrm{Mars}^{2}/2$, and final kinetic energy (just after the knock) is $$E_\mathrm{k} + \Delta{E} = m\frac{(v_\mathrm{Mars} + \Delta v)^{2}}{2}. 921