There are a number of issues to deal with here.
First of all, the volume of space you need to cover is incredibly huge. If you consider there are something like 500,000 objects orbiting the Earth alone, yet no one is particularly worried about collisions with space debris when launching a satellite, going to the ISS on a Soyuz capsule or bringing supplies up with a Dragon spacecraft (or spending months at a time on the ISS itself....). There is enough warning of oncoming satellites, used booster stages and space debris to plot orbits to avoid them or do burns that manoeuvre the spacecraft out of range.
Representation of the objects and debris orbiting the Earth
Military spacecraft will have similar or greater abilities to sense incoming space debris and manoeuvre to avoid. The only advantages you would get from that is to force the spacecraft to manoeuvre out of station and perhaps create gaps in their orbital coverage, and force them to use their manoeuvre fuel or reaction mass, and be forced to leave station to refuel and refit.
What you really want is a way to strike enemy spacecraft at long range, and with sufficient speed that they have a very low chance of evading the strike. Based on the question, we are going to ignore the idea of laser or particle beam weapons reaching out at the speed of light. A sufficiently powerful laser weapon could become a "Ravening Beam of Death" (RBoD) and capable of vapourizing metals, ceramics and carbon fibre in milliseconds at ranges of one light second (300,000Km), which would certainly be a valid way to clear a path through swarms of spacecraft far enough away that they can't fire back with kinetic energy weapons (A RBoD would need an accelerator ring @ 1Km in diameter to power an X-ray laser, so this isn't a small, swarm weapon.....) The Atomic Rockets website discusses laser weapons in the Conventional Weapons section.
What would be easier to use would be a compact weapon capable of projecting matter or energy at high velocity. This can be done through the use of nuclear devices to power weapons. Tough SF has two very good and detailed posts: The Nuclear Spear, which discusses the CASABA Howitzer, and Nuclear EFP and HEAT which discuss using the energy to drive explosively forged projectiles, streams of liquid metal or even clouds of pellets in a "shotgun" type weapon.
Theoretical reconstruction of an ORION pulse unit. Nuclear powered space weapons are derived from this basic design
Experiments done in the 1980's as part of the Strategic Demesne Initiative suggested that a small nuclear weapon could provide the energy to send a cloud of pellets at a target at velocities up to 100Km/sec. These clouds of pellets were actually meant to strip away low mass decoys and expose the warheads of an incoming ICBM strike, so the military issue is actually similar.
When you consider orbital velocity around the Earth is a "mere" 7Km/sec, this means that each pellet has vastly more energy than a simple ball bearing in earth orbit, and the enemy spacecraft has only a fraction of the time to identify the range and bearing of the incoming cloud of pellets and make a burn to evade. And this is on the "low" end of the scale. CASABA Howitzers send a spear of star hot plasma at a target at an appreciable fraction of c compounding the problems of evading the incoming fire.
So the best way to clear an area in space combat is to use nuclear powered "shotgun" shells to strip away small spacecraft by striking them hard and fast enough to prevent them from manoeuvring out of the way. "Shotgun" rounds will fill enough volume of space that the pellets that don't directly strike the initial target will remain very dangerous to any ships close by, but have the added advantage that they are moving af faster than solar escape velocity, so the firing units don't have to worry about them remaining in orbit once they have passed through the target area.
Nuclear shotgun rounds are relatively small and cheap compared to RBoD class lasers, and don't require the launching spaceship to need mirrors, cooling systems or radiator arrays, just launching racks or cells to house the missiles, or large mass drivers to shoot the warheads in the direction of the enemy. Large ships can carry more warheads, larger missiles or bigger mass drivers, so will have an advantage over small ships, even if the smaller ships can carry the same weapons. Large ships will have range advantages and overwhelm smaller ships with sheer volume of fire).