Jån löus, an kåg eta (used löba instead of wuljo, because the latter is modal)
John loves to eat cake
Subordinate clause introduced by particle/thingamajig an, which is roughly equivalent to English to in this case.
Jån wuljes, Martens etäs zen kåg (in this case it is not a modal)
John wants that Martin eat the cake
Jån wuljes, an Martens eta zen kåg
John wants that Martin to eat the cake
The latter is considered «wrong» by prescriptivists, but is found aplenty; the former uses the verb in present subjunctive, though different verbs and situations require present indicative, past indicative or past subjunctive. In both cases, Marten is in the genitive case, Martens. It derives from an older alternative, using a nominal form of the verb, such as an English gerund, but which has long fallen in disuse, roughly John wants Martin's eating [of] the cake. The first construction predates the latter.
Jån räges Martens, an ärlig läbba
John asked Martin to leave early
This is the only construction for this and is ambiguous when isolated, as there is no indication as to what the agent of läbba is. (Martens in this sentence is dative, and is just the indirect object of verb räkja, to ask, request, nothing to do with agent-genitive).