The effect of thermal radiation on the dynamics of a thermal explosion of a flammable gas mixture with the addition of volatile fuel droplets is studied. This is based on an original physical model of self-ignition. The thermal radiation energy exchange between the evaporating surface of the fuel droplets and burning gas is described using the P-1 model with Marshak boundary conditions. The original system of equations describing the effects of heating, evaporation and the combustion of fuel droplets is simplified to enable their analysis using asymptotic methods. The mathematical formulation is eventually reduced to a singularly perturbed system of ordinary differential equations. This allows us to apply the advanced geometric asymptotic technique (integral manifold method) for the qualitative analysis of the behaviour of the solution. Possible types of dynamic behaviour of the system are classified and parametric regions of their existence are determined analytically. The main attention is concentrated on the situations where delays might occur before the final ignition. Our study is focused on the impact of thermal radiation on the delay time. The dimensionless parameter responsible for the impact of thermal radiation is singled out and analysed. The dependence of the delay characteristics on the physical parameters of the problem under consideration is analysed. An explicit expression for the minimum time delay of the thermal explosion of fuel droplets in the presence of thermal radiation is derived and applied to the thermal explosion of n-decane and tetralin droplets. It is pointed out that the effects of thermal radiation can be significant, especially at high temperatures, and cannot be ignored in the analysis of this phenomenon.