  Entanglement has shown promise in enhancing information processing tasks in a
sensor network, via distributed quantum sensing protocols. As noise is
ubiquitous in sensor networks, error correction schemes based on Gottesman,
Kitaev and Preskill (GKP) states are required to enhance the performance, as
shown in [New J. Phys. 22, 022001 (2020)] assuming homogeneous noise among
sensors and perfect GKP states. Here, we extend the analyses of performance
enhancement to finite squeezed GKP states in a heterogeneous noise model. To
begin with, we study different concatenation schemes of GKP-two-mode-squeezing
codes. While traditional sequential concatenation schemes in previous works do
improve the suppression of noise, we propose a balanced concatenation scheme
that outperforms the sequential scheme in presence of finite GKP squeezing. We
then apply these results to two specific tasks in distributed quantum sensing
-- parameter estimation and hypothesis testing -- to understand the trade-off
between imperfect squeezing and performance. For the former task, we consider
an energy-constrained scenario and provide an optimal way to distribute the
energy of the finite squeezed GKP states among the sensors. For the latter
task, we show that the error probability can still be drastically lowered via
concatenation of realistic finite squeezed GKP codes.
