In general the Gaussian path integral over the $p$ momenta becomes a path integral measure density $\rho[q]$ in the Lagrangian path integral

$$\begin{align} Z[J] ~=~& \int \!{\cal D}\frac{q}{\sqrt{\hbar}}~\rho[q]~\exp\left[\frac{i}{\hbar} \left( S[q] +\int\!dt J(t)q(t) \right)\right] \cr ~=~& \int \!{\cal D}\frac{q}{\sqrt{\hbar}}\exp\left[\frac{i}{\hbar} \left( S[q]+\frac{\hbar}{i}\ln \rho[q]+\int\!dt J(t)q(t) \right)\right].\end{align}$$

The path integral measure density $\rho[q]$ may in turn be viewed as a quantum correction to the Lagrangian action $S[q]$ in the configuration space.

In general the Gaussian path integral over the $p$ momenta becomes a path integral measure density $\rho[q]$ in the Lagrangian path integral

$$\begin{align} Z[J] ~=~& \int \!{\cal D}\frac{q}{\sqrt{\hbar}}~\rho[q]~\exp\left[\frac{i}{\hbar} \left( S[q] +\int\!dt J(t)q(t) \right)\right] \cr ~=~& \int \!{\cal D}\frac{q}{\sqrt{\hbar}}\exp\left[\frac{i}{\hbar} \left( S[q]+\frac{\hbar}{i}\ln \rho[q]+\int\!dt J(t)q(t) \right)\right].\end{align}$$

The path integral measure density $\rho[q]$ may in turn be viewed as a quantum/1-loop correction to the Lagrangian action $S[q]$ in the configuration space.

Concerning field redefinitions, see e.g. this related Phys.SE post.

In general the Gaussian path integral over the $p$ momenta becomes a path integral measure density $\rho[q]$ in the Lagrangian path integral

$$\begin{align} Z[J] ~=~& \int {\cal D}q~\rho[q]~\exp\left[\frac{i}{\hbar} \left( S[q] +\int\!dt J(t)q(t) \right)\right] \cr ~=~& \int {\cal D}q~~\exp\left[\frac{i}{\hbar} \left( S[q]+\frac{\hbar}{i}\ln \rho[q]+\int\!dt J(t)q(t) \right)\right].\end{align}$$

The path integral measure density $\rho[q]$ may in turn be viewed as a quantum correction to the Lagrangian action $S[q]$ in the configuration space.

In general the Gaussian path integral over the $p$ momenta becomes a path integral measure density $\rho[q]$ in the Lagrangian path integral

$$\begin{align} Z[J] ~=~& \int \!{\cal D}\frac{q}{\sqrt{\hbar}}~\rho[q]~\exp\left[\frac{i}{\hbar} \left( S[q] +\int\!dt J(t)q(t) \right)\right] \cr ~=~& \int \!{\cal D}\frac{q}{\sqrt{\hbar}}\exp\left[\frac{i}{\hbar} \left( S[q]+\frac{\hbar}{i}\ln \rho[q]+\int\!dt J(t)q(t) \right)\right].\end{align}$$

The path integral measure density $\rho[q]$ may in turn be viewed as a quantum correction to the Lagrangian action $S[q]$ in the configuration space.