For any one interested in this type of shifted measure for fields with short-range correlation, we made some progress here in the article "Inverse of the Gaussian multiplicative chaos: Moments".
Basically, start with decomposing the variable $T_{a}$ and take supremum
$$E[\mu(T_{a},T_{a}+t)]\leq \sum_{k}E[\sup_{T\in I_{k}}\mu(T,T+t)\chi\{T_{a}\in I_{k}\}]$$
for some partition intervals $I_{k}=[a_{k},a_{k+1}]$. Next one either uses Holder inequality (or FKG inequality) to separate the two factors or a kind of strong-Markov decoupling that is true for fields with short-range correlation.
The factor $E[\sup_{T\in I_{k}}\mu(T,T+t)]$ can be studied using a chaining argument to reduce the supremum to maximum over finitely many intervals.
The factor $T_{a}\in I_{k}$ can be studied using single point estimates eg. for large $a_k$, we can use $\{T_{a}\geq a_{k+1}\}= \{a\geq \mu(a_{k+1}))\}$ and negative moments of $\mu$.