Specifically you can do two things: 1) planned incremental improvements, 2) simpler designs.
For 1), write down the entire manual workflow. Start automating pieces that are easy to automate, even if someone has to run the automation manually. Continue to automate the in-between/manual pieces. For this you can use autonomation to fall back to manual work if complete automation is too difficult/risky.
For 2), look at your system's design. See where the design/tools/implementation/etc limit the ability to easily automate. To replace a given workflow section, you can a) replace some part of your system with a functionally-equivalent but easier to automate solution, or b) embed some new functionality/logic into that section of the system that extends and slightly abstracts the functionality, so that you can later easily replace the old system with a simpler one.
To get extra time/resources to spend on the automation, you can do a cost-benefit analysis. Record the manual processes' impact for a month, and compare this to an automated solution scaled out to 12-36 months (and the cost to automate it). Also include "costs" like time to market for deliverables and quality improvements. Business people really like charts, graphs, and cost saving estimates.
