This paper investigates the evolution of initially relativistic matter, radiation, and baryons around cosmic string seed perturbations. A detailed analysis of the linear evolution of spherical perturbations in a universe is carried out, and this formalism is used to study the evolution of perturbations around a sphere of uniform density and fixed radius, approximating a loop of cosmic string. It was found that, on scales less than a few megaparsec, the results agree with the nonrelativistic calculation of previous authors. On greater scales, there is a deviation approaching a factor of 2-3 in the perturbation mass. It is shown that a scenario with cosmic strings, hot dark matter, and a Hubble constant greater than 75 km/sec per Mpc can generally produce structure on the observed mass scales and at the appropriate time: 1 + z = about 4 for galaxies and 1 + z = about 1.5 for Abell clusters.