If a person has a specific mutation in a gene, is it more likely he has other mutations in that gene?
A priori, no! Why would it? Do you have any specific hypothesis in mind you would like to discuss?

Below are some expectations from a simple model and possible reasons for why this expectation may break down under more complicated models.

Simple model

Under a simple model (panmictic population and a few other simple assumptions), the number of mutations a given individual has in any sequence considered follows a poisson distribution. Assuming that all mutations occuring have a constance selection coefficient $s$, a constant dominance coefficient $h$ and that the mutation rate for the sequence of interest is $U$, then the number of mutations an individual carry comes from a Poisson distribution with mean $\frac{U}{2hs}$ (Crow 1970).

This model is simple but is probably a pretty good approximation to reality. Below are three assumptions that are not necessarily true and that would yield to a higher variance in the number of mutations (that is a higher probability for an individual that already carry a mutation to get a second mutation).

Population structure

In reality, populations are often not panmictic. As the genetic load depends upon the population size. More specifically, the smaller the population, the higher the genetic load (or the higher the number of mutants alleles present in the population) (Kimura et al. 1963).

Past evolutionary and demographic history

Populations are not stable through time. Some population have recently gone through a bottleneck, others a quickly expanding, some are shrinking, some have recently suffered from a plague, etc... Such processes can all cause populations to differ in the number of mutations they carry and therefore would increase the variance in the distribution of number of mutations individuals carry.

Within a specific short enough sequence, physical linkage associated with specific past evolutionary process will create a relatively long lasting linkage disequilibrium.

Condition-dependent mutation rate

It has been shown in Drosophila melanogaster that individuals with poor genotype tend to have a higher mutation rate causing their lineage to accumulate even more mutations (Sharp and Agrawal 2012).