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(Non-uniqueness of mild solutions to supercritical heat equations. arXiv:2501.17032 (2025)), it has been shown that the focusing power nonlinearity heat equation\n \n \n $$\\begin{aligned} \\partial _t u -\\Delta u = |u|^{p-1}u, \\quad p>1, \\end{aligned}$$<\/jats:tex-math>\n \n \n \n \n \n \n \n \u2202<\/mml:mi>\n t<\/mml:mi>\n <\/mml:msub>\n u<\/mml:mi>\n -<\/mml:mo>\n \u0394<\/mml:mi>\n u<\/mml:mi>\n =<\/mml:mo>\n \n \n |<\/mml:mo>\n u<\/mml:mi>\n |<\/mml:mo>\n <\/mml:mrow>\n \n p<\/mml:mi>\n -<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n u<\/mml:mi>\n ,<\/mml:mo>\n \n p<\/mml:mi>\n ><\/mml:mo>\n 1<\/mml:mn>\n ,<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mtd>\n <\/mml:mtr>\n <\/mml:mtable>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:disp-formula>\n in dimensions\n \n \n $$d \\ge 3$$<\/jats:tex-math>\n \n \n d<\/mml:mi>\n \u2265<\/mml:mo>\n 3<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n has non-unique local solutions in\n \n \n $$L^q(\\mathbb {R}^d)$$<\/jats:tex-math>\n \n \n \n L<\/mml:mi>\n q<\/mml:mi>\n <\/mml:msup>\n \n (<\/mml:mo>\n \n \n R<\/mml:mi>\n <\/mml:mrow>\n d<\/mml:mi>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n for\n \n \n $$q < d(p-1)\/2$$<\/jats:tex-math>\n \n \n q<\/mml:mi>\n <<\/mml:mo>\n d<\/mml:mi>\n (<\/mml:mo>\n p<\/mml:mi>\n -<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n \/<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n provided that\n \n \n $$p < p_{JL}$$<\/jats:tex-math>\n \n \n p<\/mml:mi>\n <<\/mml:mo>\n \n p<\/mml:mi>\n \n JL<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , where\n \n \n $$p_{JL}$$<\/jats:tex-math>\n \n \n p<\/mml:mi>\n \n JL<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n denotes the Joseph-Lundgren exponent. In this paper we investigate the effect of different randomizations on the well-posedness of the equation. First we show that adding a forcing term white in time and colored in space in (NLH) is not sufficient to improve the solution theory: namely, we prove non-uniqueness for local-in-time mild solutions of (NLH) with additive noise. 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