How do I account for temporal autocorrelation?

Question

I am trying to fit a beta regression model using GAMLSS.

The data:

For each $y$, we have an indication of what patient $p$ it is, an indication of what month $m$ the observation took place, and at what location $l_1, l_2$ (longitude and latitude) the observation took place. We then further have covariates such as sex.

Here's an example of what the data set might look like:

y   p  m   l_1 l_2 sex
0.5 1 Jan  22  -49  M
0.7 1 Jan  25  -51  M 
0.4 1 Feb  21  -52  M 

0.5 2 Jan  21  -51.5  F
0.5 2 Jan  24  -51   F 
0.5 2 Jan  21  -51  F
0.5 2 Feb  20  -52  F 
0.5 2 Feb  21.5  -55  F    

The model:

This is my question. How should I set up the model to account for autocorrelation, both temporal and spatial?

I know that I can model the longitude and latitude using a smooth function $f(l_1, l_2)$, so that the model might be

gamlss(y ~ sex + f(l_1,l_2) + random(p), family = BE)

where $p$ is a random effect for each patient, and sex is a fixed factor effect.

For temporal autocorrelation, is it enough to include it as a fixed effect?

gamlss(y ~ sex + m + f(l_1, l_2) + ranndom(p), family = BE)?