Problem with display of presentation

You asked,

How do I get rid of ... the "University of Lincoln" bit [in the banner line]?

You need to change

\institute{University of Lincoln School of Maths and Physics}

to

\institute[]{University of Lincoln School of Maths and Physics}

By default, the "short" version of the institute-related information is equal to the material in curly braces, i.e., the main argument of \institute. The optional short version is what's placed in square brackets. By placing nothing inside the square brackets, nothing gets passed to the banner line shown at the bottom of the page.

The following screenshot shows the banner line once the document is recompiled:

The Madrid theme uses the infolines outer theme in which the contents of the footline are defined. In the original definition (copied from beamerouterthemeinfolines.sty you can find the following line:

\begin{beamercolorbox}[wd=.333333\paperwidth,ht=2.25ex,dp=1ex,center]{author in head/foot}% \usebeamerfont{author in head/foot}\insertshortauthor\expandafter\ifblank\expandafter{\beamer@shortinstitute}{}{~~(\insertshortinstitute)}

that inserts the author name and the institute's name into the leftmost colored box in the footline.

In order to remove the institute's name you can redefine the footline as follows:

\documentclass[11pt]{beamer}
\usetheme{Madrid}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage[english]{babel}
\usepackage{graphicx}
\author{Anthony Robinson}
\title{Irrational and Transcendental Numbers}
\subtitle{Mathematics Project}
%\logo{}
\institute{University of Lincoln School of Maths and Physics}
\date{April 30, 2019}
%\subject{}
%\setbeamercovered{transparent}
%\setbeamertemplate{navigation symbols}{}

\makeatletter
\setbeamertemplate{footline}
{
  \leavevmode%
 \hbox{%
  \begin{beamercolorbox}[wd=.333333\paperwidth,ht=2.25ex,dp=1ex,center]{author in head/foot}%
    \usebeamerfont{author in head/foot}\insertshortauthor\expandafter
  \end{beamercolorbox}%
  \begin{beamercolorbox}[wd=.333333\paperwidth,ht=2.25ex,dp=1ex,center]{title in head/foot}%
    \usebeamerfont{title in head/foot}\insertshorttitle
  \end{beamercolorbox}%
  \begin{beamercolorbox}[wd=.333333\paperwidth,ht=2.25ex,dp=1ex,right]{date in head/foot}%
    \usebeamerfont{date in head/foot}\insertshortdate{}\hspace*{2em}
    \usebeamertemplate{page number in head/foot}\hspace*{2ex} 
  \end{beamercolorbox}}%
  \vskip0pt%
}
\makeatother

\begin{document}

    \maketitle


\end{document}