Issue with LaTeX code in "header-includes" in R Markdown file after R and RStudio upgrade

Welcome to TeX.SE. I changed the header-includes syntax and put the definition of the microtype macro, MT@is@composite, into a file called mystyles.sty and then just added mystyles.sty to the header-includes section. All those extra packages you are loading, and definecolor and pagenumber commands could be added to mystyles.sty if you wanted to do that. You might have to revert my change to setmainfont since I changed this to what worked for me on a Windows system.

---
output:
  pdf_document:
    keep_tex: true
    latex_engine: xelatex
  html_document:
    df_print: paged
header-includes:
  - usepackage{microtype}
  - usepackage{fontspec}
  - setmainfont{Times New Roman} #<- changed this
  - usepackage{ragged2e}
  - renewcommand{footnotesize}{scriptsizejustify}
  - usepackage{setspace}
  - usepackage{xcolor}
  - definecolor{very-light-gray}{gray}{0.95}
  - pagenumbering{gobble}
  - input{mystyles.sty}
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE)
```

## R Markdown

This is an R Markdown document. Markdown is a simple formatting syntax for authoring HTML, PDF, and MS Word documents. For more details on using R Markdown see <http://rmarkdown.rstudio.com>.

When you click the **Knit** button a document will be generated that includes both content as well as the output of any embedded R code chunks within the document. You can embed an R code chunk like this:

```{r cars}
summary(cars)
```

## Including Plots

You can also embed plots, for example:

```{r pressure, echo=FALSE}
plot(pressure)
```

Note that the `echo = FALSE` parameter was added to the code chunk to prevent printing of the R code that generated the plot.

This is the content of mystyles.sty:

makeatletter
defMT@is@composite#1#2relax{%
  ifx#2else
    expandafterdefexpandafterMT@charexpandafter{csnameexpandafter
                    stringcsnameMT@encodingendcsname
                    MT@detokenize@n{#1}-MT@detokenize@n{#2}endcsname}%
% 3 lines added:
    ifxUnicodeEncodingName@undefinedelse
      expandafterexpandafterexpandafterMT@is@uni@compMT@chariffontcharelsefirelax
    fi
    expandafterexpandafterexpandafterMT@is@letterMT@charrelaxrelax
    ifnumMT@char@ < z@
      ifMT@xunicode
        edefMT@char{MT@exp@two@cMT@strip@prefixmeaningMT@char>relax}%
          expandafterMT@exp@two@cexpandafterMT@is@charxexpandafter
            MT@charMT@charxstringrelaxrelaxrelaxrelaxrelax
      fi
    fi
  fi
 }
% new:
 defMT@is@uni@comp#1iffontchar#2else#3firelax{%
  ifx#2elseedefMT@char{iffontchar#2fi}fi
}
makeatother

Your header content then appears in the preamble of the .tex files, just as intended. I added keep_tex: true to the YAML to keep the intermediate .tex file that is produced so that we can see what's going on in the header.

usepackage{microtype}
usepackage{fontspec}
setmainfont{Times New Roman}
usepackage{ragged2e}
renewcommand{footnotesize}{scriptsizejustify}
usepackage{setspace}
usepackage{xcolor}
definecolor{very-light-gray}{gray}{0.95}
pagenumbering{gobble}
input{mystyles.sty}

If you are having trouble, make sure you Clear Knitr Cache before reexecuting the .Rmd file.