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If "strict fibre" just means the fibre over *, then you can make the construction as follows. There are two steps. First, let f:X->X be homotopic to the identity and contract some neighborhood N of * onto *. Pulling back your fibration p through f, you are reduced to the case where p is a projection FxN->N over N. Then, consider a function g: N->I whose value is 0 on partial N and 1 on *; consider the mapping cylinder C of a homotopy equivalence F->F' and the projection pr:C->[0,1] with pr^-1(1)=F'; define E' over N as the amalgamated product of C with N over G; define E' as E over (X-N). if you have chosen N to retract by deformation on *, then this E' will have the property you want. Maybe there is a more elegant way to present the construction (i'm not an expert in homotopy theory nor cw-complexes)