Genome editing technologies that install diverse edits can widely enable genetic studies and new therapeutics. Here we develop click editing, a genome writing platform that couples the advantageous properties of DNA-dependent DNA polymerases with RNA-programmable nickases (e.g. CRISPR-Cas) to permit the installation of a range of edits including substitutions, insertions, and deletions. Click editors (CEs) leverage the "click"-like bioconjugation ability of HUH endonucleases (HUHes) with single stranded DNA substrates to covalently tether "click DNA" (clkDNA) templates encoding user-specifiable edits at targeted genomic loci. Through iterative optimization of the modular components of CEs (DNA polymerase and HUHe orthologs, architectural modifications, etc.) and their clkDNAs (template configurations, repair evading substitutions, etc.), we demonstrate the ability to install precise genome edits with minimal indels and no unwanted byproduct insertions. Since clkDNAs can be ordered as simple DNA oligonucleotides for cents per base, it is possible to screen many different clkDNA parameters rapidly and inexpensively to maximize edit efficiency. Together, click editing is a precise and highly versatile platform for modifying genomes with a simple workflow and broad utility across diverse biological applications.