Electronic strucure of cyclopropene radical cation Suppose I have a cyclopropene molecule with a + charge on the singly bonded carbon.I think the singly bonded carbon is $\mathrm{sp^2}$ hybridised, isn't it? To obtain this species from cyclopropene, one can consider removing two hydrogen atoms, one homolytically (the one bonded to pure p orbital of carbon) and the other heterolytically (the one bonded with $\mathrm{sp^2}$ orbital of carbon).

So, finally, there is one electron in the pure p orbital and one unit positive charge on one $\mathrm{sp^2}$ hybridised orbital, shouldn't the electron in the pure p shift to the $\mathrm{sp^2}$ orbital as $\mathrm{sp^2}$ has more s character (so it is near the nucleus), does this shift happens?

I am unclear about this scenario, can someone please explain where the electron and the positive charge reside in the C atom.
You do not really have a singly bonded carbon atom. When you remove the hydrogens from the originally saturated carbon the pi electrons delocalize to make an aromatic ring.  In effect you have a cyclopropenyl cation with one of the hydrogen atoms homolytically removed.  The positive charge is thus delocalized in the ring but the odd electron is localized to the carbon with no hydrogen, in the outward pointing $sp^2$ hybridized orbital.