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We focus on the spreading properties of solutions of monostable reaction–diffusion equations. Initial data are assumed to have heavy tails, which tends to accelerate the invasion phenomenon. On the other hand, the nonlinearity involves a weak Allee effect, which tends to slow down the process. We study the balance between the two effects. For algebraic tails, we prove the exact separation between ‘no acceleration’ and ’acceleration’. This implies in particular that, for tails exponentially unbounded but lighter than algebraic, acceleration never occurs in the presence of an Allee effect. This is in sharp contrast with the KPP situation [20]. When algebraic tails lead to acceleration despite the Allee effect, we also give an accurate estimate of the position of the level sets. [ABSTRACT FROM AUTHOR]