Fit the model

formula <- y ~ 0 + b0 + f(s, model = spde)

res <- inla(
  formula,
  family = 'poisson',
  data = inla.stack.data(stk.full),
  control.predictor = list(
    compute = TRUE,
    link = 1,
    A = inla.stack.A(stk.full)
  ),
  E = inla.stack.data(stk.full)$e
)

E is the offset: the known component in the mean for the Poisson likelihoods, defined as \(E_i \cdot e^{\eta_i}\).